Syntaxis Mathematica: OR, A CONSTRUCTION Of the harder Problems of GEOMETRY: With so much of the conics as is therefore requisite, And other more ordinary and useful PROPOSITIONS intermixed: And TABLES to several purposes. The Contents follow in the ensuing leaf. By THO. GIBSON. Virgil. Tempora dispensant usus & tempora cultus. LONDON, Pridted by R. & W. Leybourn, for Andrew Crook, and are to be sold at his Shop at the Green Dragon in St. Paul's Churchyard. 1655. CONTENTS. Chap. 1. Explanation of the Symbols. Page 18 Addition, Subtraction, Page 22 Multiplication, Division. Page 23 Chap. 2. Of Aequations. Page 25 Rules to find the Roots in the three sorts of Quadratique Aequations. Page 26 Demonstrations thereof. Page 27 Reduction of Aequations. Page 29 Chap. 3. Resolution of Aequations according to the general method. Page 32 Resolution of Mixed cubics. Page 40 Chap. 4. Of the Genesis of Aequations, number of Roonts, changing them by Addition, Subtraction, Multiplication and Division. Page 55 Chap. 5. Reduction of Solids to one sum. Page 69 Chap. 6. Of Surd Numbers. Page 75 Chap. 7. Solution of divers Problems Page 82 Chap. 8. Of mixture, of Metals and Liquors, divers Propositions and Rules. Page 94 Chap. 9 Of Mensuration, the superficies and solidity of divers Bodies, and fragments of a Sphere. Page 112 Chap. 10. To find an Ellipsis in any proportion. Page 124 Chap. 11. Definition general of a Cone, Definitions and descriptions of the Sections, to draw Tangents, Asymptots, and to find out the Burning points, Centres, and other things. Page 130 Chap. 14. To find two means, trisect an arch, extract the Root of any cubical or Biquadraticall Aequation, not proposed in numbers, by a portion of a Parabola. Page 154 Chap. 15. Two Geometrical Problems. Page 168 Chap. 16. Of Dialling upon any Plain, with demonstrations of the hardest Page 180 Chap. 17. Of Azimuths, Almicanters, Jewish Hours, etc. Page 197 Chap. 18. What Section distinguisheth the light and shadow, at any time on any plane. Page 212 A Table of semidiurnal arches, for 32 Latitudes. Page 220 A Table of the Amplitudes, for 27 Elevations. Page 221 A Table of Houre-arches, for 21 Elevations. Page 222 The making these Tables, and for right Ascension and Obliqne, ascensional difference, etc. Page 223 Two Tables of compound Interest and Rents from 5 to 10 per Cent. for 31 years. Page 233 TO THE READER ALthough no Book can be so copious, as wherein every Reader may be furnished with every thing which he looks for, yet there is seldom any thing like a Book which may not afford the Reader something which he looked not for. The hope that I may do so, may be taken for the reason why I writ this. But why I writ thus, that is, Analytically, Valesius, Lib. de Philosoph. Sacr. pag. 8. shall answer for me. Si quis velit resolutivum tenere ordinem, qui discentium naturae magis se accommodat, & petitionibus minus indiget, quia incipit à postremis de quibus primùm omnium continget dubitare. That is, If any one would hold the Resolutive order, which more accommodates its self to the nature of Learners, and less needs Petitions, because it gins from the last things, of which principally men happen to doubt. The method here used is the same as in Master Harriot in some places, that is, in such Aequations as are proposed in numbers. And as in Des Cartes in some other places, that is, in such Aequations as are Solid, and not in numbers. Not that the Book is taken out of them, much less that it proceeds continually with them, but disjunctly, as I thought fit to intermix them among other things which are not in them. I shall use no arguments to commend the Mathematics, or prefer them before the Dogmatiques, for this is but to write in praise of Hercules. Yet this may be said of them, that although some bodily exercises conduce more to health, and some mental labours more to wealth, yet nothing affords the mind more pleasure, or more profit, with less repentance. And therefore in a dull solitude, or vacancy of business (both which may happen to Gentlemen) these are amiable company, which yield a delightful and innocent expense of leisure. As for that Question (which is frequent) What profit is in these hard Studies? it needs no answer, because it imports the ignorance or idleness of the Asker, or rather both. For first (and which may be reprehensive to many Writers, that must not be called Authors, which of late have brought up a fashion to write in a Querying way) he declares his ignorance, otherwise he needs not ask that which he knows. Secondly, If some Meats were recommended to a man, with which he is yet unacquainted (otherwise they needed not recommendation) if he should first ask whether they be good or not, that is, whether they would please his and stomach? the Question is absurd, for he cannot know that until he have tasted and digested. So the idleness of the asker may from hence be discovered. Nor is there any profit to be gained by any Science, except the Science be first gained by industry. Besides, to think others, who being once entered herein, should delight so much in them, as to make them a study all their life, if there were no profit in them; or if it were so, nevertheless to recommend them to others, signifies another bad quality or two, which I forbear to name. As for the difficulty of these Sciences, I must confess that the first Aspect of them may seem uncouth and horrid (Radices doctrinae amarae sunt, fructus tamen dulces) yet there is no reason why he should be deterred hereby, and not rather animated with desire to go as far as another, or else with shame to think there should be so many Books in the world, easy to others and useful, but to him not understood, and therefore useless. In this following Treatise, my chief care hath been to render it all intelligible to every Reader, and I doubt not but it will prove so to every diligent one. The Symbols and Characters herein used, are such as have been long accepted in the world, without any innovation or fancy of my own, for although every Writer hath equal liberty herein, to add or alter, as he sees (or rather thinks) fit, yet in my opinion, we ought not to do this without considerable cause, or a kind of convenience equipollent to necessity: for without doubt, he that increaseth these, increaseth his Readers burden, especially if such increase be needless. I leave the rest to the Reader to censure as he finds cause, and it is in vain to do otherwise, for in these demonstrable things, the Readers detection of any error (of judgement) will be acceptable even to him that writ it, if he be civilly acquainted with it, but the said Readers detraction cannot here hurt any one but the Reader. T. G. ERRATA. PAge 68 line 2. read eee = ccc. p. 70. l. 20. r. cccc ³ = 729. p. 76. l. 21. r. 3 √ 5 = √ 45. p. 96. l. 26. r. Proposition. p. 113 at Section 4. deal Eucl. 12.7. and at Section 5. writ Eucl. 12.7. p. 124 & p. 129. in the Diagram the letters x and y are to be supplied upon the Diameter ac, see p. 128. l. 19 p. 134. l. 16. r. Vmbilicus. p. 138. l. 8. r. fk, go, tl, etc. p. 151. l. 1. r. z = 132. p. 178. l. 28. r. for practise. p. 183. l. 1. r. right angles. p. 185. the same diagram as in p. 183. should be used. p. 187. l. 13. r. Circle wne. & l. 14. r. ns. p. 199. l. 7. deal of. p. 203. l. 1. deal the. p. 205. l. 17. r. the arch fb. Preface. ALthough in general the Mathematics, and especially the Analitiques, are easier in the beginning then proceeding (for the hardest is reserved for last) the Principles, Petitions, and Definitions also, seldom meeting any opposition, being (for the most) first sight lessons to all: yet I have thought fit, for some men's sake (who expect it in all Books) to premise some initiary things so easy, and so well known already, as must be received by every one. Nevertheless, that it may not seem trifling to the already knowing party, I will not be ample. Common Sentences. I. Greater. 1 The whole then the part. 2 Equal to that which is greater. 3 Greater then that which is greater. 4 Neither less nor equal. 5 Double of the whole. 5 Multiplex of the whole. 6 Greater than that which is equal, Euclid 1.16. 7 Any thing commune or equal added to greater or to less, is greater than it. Euclid 4.17. 8 Where the parts are greater than the parts, the whole is greater than the whole. 9 Of two things, that which hath greater proportion to a third. II. Less. 1 The part then the whole 2 Equal to the less. 3 Less than the less. 4 Neither greater nor equal. And so by a way contrary to the former may be form all that's Less. III. Equal things. 1 That which is common to two others, is equal to itself, as in Euclid 1.5. an angle is common, and in Euclid 1.7.8.9.10.11.12. a side is commune. 2 Those which are equal to the same thing. 3 Which are equal to equal things. 4 Which are equal to nothing. 5 The whole of equal things, added to equal. Ax. 2. 6 The remain of equal things taken from equal things. 7 The whole of equal things added to a common thing. Or contra. Eucl. 1.6.9.10.11.12. 8 The remain of equal things when a common thing is deducted. 9 Vertical angles. 10 The Rectangles of the Means and Extremes. 11 Things which agree among themselves. Ax. 8. this last is proper to Geometry. 12 That which is not unequal, that is neither greater nor less: this is proper to homogeneals, for heterogeneals admit no comparison. 13 The whole to all the parts together. 14 The halves of the whole. 15 Whose halves are equull. 16 Whose parts are equal in Number and Magnitude. 17 Whose Doubles are equal. 17 Whose Equimultiplices are equal. 18 If the parts be equal to the parts, the whole is equal to the whole. 19 If nothing else be equal besides the thing supposed, that thing is equal. 20 Which have the same proportion to the same thing. 21 Those to which the same thing hath the same proportion. 22 Of four proportionals, if the first be equal to the third, the second is equal to the fourth. 23 If there be twice three Magnitudes, which taken by two and two are in the same proportion, if (of equality) the first be equal to the third, the fourth is equal to the sixth. Euclid 5.20. 24 If there be twice three Magnitudes, which taken by two and two, are in the same proportion, and the proportion be perturbate, if the first be equal to the third, the fourth is equal to the sixth. Euclid 5.21. iv Agreeing things. 1 Are such as are equal, and of the same kind. V, Unequal things. 1 Greater or less, 2 The whole and the part. 3 The whole, when a common thing is added to unequal things. 4 The whole, when an equal thing is added to unequals. Euclid 1. Ax. 4. 5 The Remain, when a common or equal thing is taken from unequal things. VI Double. 1 The double of the half. 2 Two equal things taken together are double to one of them. 3 The double of that which is equal. 4 That which is equal to the double, 5 If the parts be double to the parts, the whole is double to the whole. The 6 proportion of like figures to their sides of like proportion, Euclid 6.19. VII. Halfe. 1 Is the half of that which is double. 2 That which is equal to one of two equals, is the half of them together. 3 The half of an equal thing. 4 That which is equal to the half. 5 The proportion of like sides to the proportion of like figures. VIII. A thing is. 1 If nothing else which can be proposed is the thing, than this which was proposed is. Or, 2 If any thing else besides that supposed be put, and an impossibility follows, than this which was supposed is that which was sought. 3 If that which is supposed be nothing else, than it is what was required. Or, 4 If this which is supposed being put for any thing else, an impossibility follow, than it is what was required. 5 That which necessarily follows from that which is. 6 Which put for not in being, there follows an impossibility. IX. Something. 1 Is that which if any thing be added to it, it is more, or if any thing be taken from it, it is less, or to which if nothing be added, it is the same. 2 To which if less than nothing be added it is less. 3 If less than nothing be substracted, it is more. 4 Which multiplied by something is more. 5 Which multiplied by nothing is nothing. 6 Which multiplied by less than nothing, is less than nothing. 6 Or that which divided by Something, is less. Nothing, is nothing. Less than nothing, is less than nothing. X. Nothing. 1 Is that which added to, or taken from something, or less than nothing, leaves it the same it was: and multiplying or dividing something produceth nothing, but takes the thing quite away. XI. Less than nothing. 1 Is that which added to something makes it less. 2 Which substracted from something makes it more. 3 Which added to less than nothing makes it still less. 4 Which taken from less than nothing makes it more. 5 Which multiplied by something gives less than nothing. 6 Which multiplied by less than nothing, produceth something. 7 Which divided by something, makes less than nothing. 8 Which divided by less than nothing, makes something. XII. Unity. 1 Is that to which if unity be added it is doubled. 2 From which if unity be taken, it is nothing. 3 If more than unity be taken, it is less than nothing. 4 If less than unity be taken, it is less than unity. 5 If less than nothing be taken, it is more than unity. 6 Is that which cannot be multiplied or divided by unity without remaining the same. 7 Is the difference of the two greater sides of a rectiline rectangle triangle: or may be so by reduction of the sides to lesser numbers. vide Corol: ad Cap. 7 Prob. 3. Propositions of EUCLID, fit to be known to the ANALIST. In the first Book Prop. 6, 13, 14, 15, 18, 19, 28, 32, 43, 47, 48. In all 11. In the second book all but the eleventh, and last, in all 12. In the third, Prop. 14, 20, 22, 31, 32, 35, 36, in all 7. In the fift Book, Prop. 15, 16, 17, 19, 24, 25, in all 6. In the sixth Book, Prop. 2, 3, 4, 6, 7, 8, 13, 14, 16, 19, 24, 31, in all 12. Many more propositions out of these and the remaining Books might be useful: But these 48 last reckoned are such as (in my judgement) ought chief to be read, and remembered, for assisting to attain and resolve Aequations. Now whereas it is said in the ensuing Chapter, that vowels are put for things unknown, or sought; and consonants ever for known things, it is to be noted that in a Scheme which employeth almost all the Alphabet these are promiscuous. But in abbreviation or demonstration, wheresoever one single letter is put (or supposed to be) equal to any line or number, although the same letters which before designed the Diagram be, again used herein, yet in a different acception; For whereas in the Diagram they signified points. now they stand for lines or things; And evermore the consonants signify things given or known before; and the vowels (although all present) are supposed equal to things which are not yet known, but about to be found. Only the vowel o is seldom used in this sense, because it is usurped in another, that is to signify nothing. As a − b = o. signifies that a want b is equal to nothing: or that a is equal to b where the vowel o stands for a cipher, that is nothing. On the other side the Greek vowal y is usually put for any unknown quantity. Definitions. Definition I. The unknown Quantity of any aequation is called generally Potestas; or a Power, Quantity, or Term. Definition II. A Rectangle is in numbers the Product of two numbers multiplying one another. In Geometry it is the Area, space, or content of a right angled quadrangular figure, made also by multiplication of two lines, which are called the sides; of which one is the measure of the breadth the other of the length. Definition III. A Rectangled Parallelepipedon is the product of a Rectangle multiplied by a right line or number: And if that line or number and the length and breadth of the Rectangle be severally equal it is a Cube, or Die. Definition IU. A Prism is a Solid contained within five superficies of which three are Quadrangular, and the other two being opposite, are triangular: Or it is like the top of an ordinary English house cut off by a Plane passing through or parallel to the Eaves. The rest of this kind I shall not desine here but refer the Reader to Euclid. The names of the Potestates or Powers. 1 The first Power is called a Side, or Root: The later word Root is most used here; and it is signified thus, a. 2 The second Power is called a Square, and is thus written, aa. 3 The third is called a Cube, and is thus written, aaa. or sometimes for brevity, a3 4 The fourth, a Biquadrat or squared square, anciently a Zenzi zenzick, figured thus zz now thus, aaaa, or for brevity, a4. 5 The fift Power is called a Sursolid, and is written thus, aaaaa; or briefly thus a5. 6 The six, a squared Cube, or zenzicube, written thus, aaaaaa or a6. 7 The seventh, a second Sursolid, and is written aaaaaaa, or more short a7. 8 The eight is called a squared square squared, or zenzi zenzi zenzi zenzick, and is written aaaaaaaa; or thus a8, etc. Consectary I. Hence it is manifest that these powers uninterrupted, are in continual proportion, the proportion of them being as a, to unity: or the converse. Consectary II. It is also here plain, that every Power hath so many dimensions, as the letters, with which it is written. For a4 being written with four letters, if one letter stand for one dimension, that is length or breadth, the other three arise by three several Multiplications, and every Multiplication adds a dimension, in this sense. A Table of the Powers of the Digits. 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 2 4 8 16 32 64 128 256 512 3 9 27 81 243 729 2187 6561 19683 4 16 64 256 1024 4096 16384 65536 262144 5 25 125 625 3125 15625 78125 390625 1953125 6 36 216 1296 7776 46656 279936 1679616 10077696 7 49 343 2401 16807 117649 823543 5764801 40353607 8 64 512 4096 32768 262144 2097152 16777216 134217728 9 81 729 6561 59049 531441 4782969 43046721 387420489 In the former Table, the Digits at the top 2, 3, 4, etc. Show the Columns of the second, third, fourth, etc. Powers. The Digits, at the left side, show the several Roots or first Powers, and their proportion to Unity. All the rest is evident. And now because towards the end of this little Treatise, I shall happen to speak once ortwice of Arithmetical Calculation; The Reader may hereby understand, that such Calculations are usually (and most easily) performed by numbers assumed in Arithmetical proportion, called Logarithmes; of which I intent to say nothing, supposing any Reader conversant about such things wherein I use them, cannot be ignorant of them and their use. Such as be, should read that first, of which they need no better (nor other) instructions than such as they may have in Mr. Norwoods' doctrine of Triangles; which is a Book not very dear. But to such as have not that, these following directions may be of some use. 1 In every Spherical triangle which hath one right angle, or one side a quadrant, all the other five parts (for every triangle hath six in all, that is three sides, and three angles) are called Circular Parts. 2 Of these Circular parts, if any two be given, the rest, that is, any one of the rest, may be found ta one operation. 3 For those two are either adjacent, or remote, or opposite: if adjacent, & the part required be also adjacent (or touching) to one or either of them, then that one so touched on the one side by a part given, and on the other side by a part required, may fitly be called The Middle Part And it is a demonstrated truth, that, As the tangent of the known part adjacent, is to the right sine of the middle part; So is the Radius or Semidiameter, to the tangent of the unknown, or required part; being also adjacent to the middle part, as before. And therefore. if instead of the natural sins and tangents, the Logarithmes be used, they being in Arithmetical proportion, the sum of the two middle terms is equal to the sum of the two extremes; And so here, the fine of the middle part plus Radius is equal to the tangent of the of the adjacent part known plus the tangent of the part required. I hope the word plus needs no interpretation. Note 1. It is notwithstanding to be ever remembered that every of the five circular parts must be considered two ways; that is whether it be contiguous to the right angle, or quadrant; if so then all before is right and unalterable. But if not so, that is, if some other part lie betwixt them, than all that hath been said of their sins and tangents, which were then supposed contiguous, must be performed by the sins compliments and tangents of the compliments of such of the parts respectively as are remote from the right angle or quadrant. 4 If now the two parts given be remote, and the part required lie betwixt them, then make the part required the middle part, and it may be found as easily as in the former case. 5 If the two known parts be contiguous, and the part required adjacent to neither of them, but opposite to one of them, than (working still by Logarithmes) make the part required the middle part, and then the sine of the middle part plus Radius, is equal to the sins compliments of the opposite parts given, if therefore from those two sins compliments added, be taken Radius, the rest is the sine of the thing required. Note 2. It is further to be noted that the sins compliments of those parts which by the former note are accounted compliments, are the sins of the things themselves. Example. In the triangle zps let p be the Pole, s the Sun and z the Zenith zps, the hour from noon in Winter, or the hour from Midnight in Summer; pzs, the Azimuth from the North; and zsp, the angle of position: and sz be 90 Deg. as at Sun-rise. mathematical diagram Then first, sine z plus Radius, is equal to tangent compliment zp plus tang. s. Secondly, sine compl. p plus Radius is equal to tang. compl. pz plus tang. compl. ps. per Not. 1. Lastly, (by the 5 direction) sine compl. sp plus Radius is equal to the sine of pz plus the sine compliment of z; because pz is accounted a compl. and z not so. Note 1. And if instead of a quadrant (as zes) there were a right angle, at p, or at z, or at s, the aforegoing directions serve. If there be neither right angle nor quadrant, there must be two operations to do this, having first supposed a circle to pass from one of the angles to cut the opposite side (produced when need is) at right angles. This perpendicular circle shall fall sometimes within the triangle, sometimes without. Within, when the other angles are both obtuse: Or both acute, as at the hours between six and noon are the angles at p and s. In letting fall the perpendicular, Mr. Norwoods' advice is to do it. 1 From the end of a side given, being adjacent to an angle given, let it fall opposite to that angle. 2 And touching in some part the side required. 3 And opposite (if it may be) to the angle required. One of the most difficult cases in obliqne angled spherical triangles is this. mathematical diagram In the triangle pzs. Let there be given, comp. Elevation pz = 38. 28′ comp. Declinati. ps = 66. 29′ comp. Azimuth pze = 70. 0′ To find the compliment of the Sun's Altitude zs = 75. 45′ having produced sz and from p, let fall poe, perpendicular to it: then, First, making pze (the middle part) s.c. pze. plus Radius 195340517 From which taking the tangent of the Elevation, t.c. zp. 100999135 Remains tang. of ze = 15. 12′t. ze 94343382 Secondly, to sine compl. ze 99845347 Add sine of Declination, s.c. ps 96009901 The sum of them is 195855248 From which taking sine Eleva. s.c. pz 98937412 Remains (the sine of 29. 27′) s.c. 60. 33′ 96917696 To whose comple. 60. 33′ adding ze = 15. 12′, the sum is 75. 45′ the thing required. Whereas, if the Azimuth itself pzs were 70d, then ze being taken from 60. 33′, the rest 45. 21′ is the Sun's Altitude. CHAP. I. An explanation of the Characters and Symbols used in this Work. FIrst, One single letter of the Alphabet is usually put for any quantity whatsoever, as well Line as Number, whether known or unknown. But for the most part, where any quantity is sought, there a or some other Vowel is put for it; and the other Quantities known, are signified by Consonants. These letters are multiplied one into another by joining them together without any prick or comma between, nor doth it import at all which is first or last written: for bcd, bdc, and cbd; are all one. So a multiplied by a produceth aa. And a multiplied by b, produceth ab. And ab multiplied by c, produceth abc. The like of all others whatsoever, except fractional quantities; as, ab+ fg/ d and bc − fb+ rc/ b+ c If the first of these were to be multiplied by d, it is done by taking away the d under the line, and the product is ab + fg. If the second were to be multiplied by b + c, it is done by taking away the Denominator b + c, and the product will be bc − fb + rc. For all Fractions aswell in Plain as in Figurate Arithmetic, are nothing else but Quotients of one number divided by another; and are multiplied again by taking away their Divisor, and line of Separation. Division is done in Figurative Arithmetic, most commonly by applying some line of separation between the Dividend and the Divisor. So a/ b is a divided by b, and abc/ f signifies that abc is divided by f. But yet if the letter f had been found in the Dividend, the application of this line had not been necessary, for it might have been better done by taking away that letter out of the Dividend. So afc divided by f quotient is ac and ffcc divided by fc quotient is fc by ff quotient is cc by cc quotient is ff by ffc quotient is c by fcc quotient is f by f quotient is fcc by c quotient is ffc And the like may easily be understood of all the rest. Symbols of Majority > Minority < Equality = Addition + Substraction − Root of a quantity √ Proportionality continued ′ ″ ‴ ' ' ' ' Proportionality disjunct ′ ″ ′ ″ So b > c signifies b greater than c b < c .. b less than c b = c ....... b equal to c b + c ....... c added to b b − c .. c taken from b √ 72 signifies the square root of 72, etc. And b′ c″ d‴ f ' ' ' ' signifieth that as b is to c, so is c to d, and so d to f. Likewise b′ c″ f′ g″ signifies that as b is to c, so is f to g. These things before expressed are almost generally received: and used not only for brevity in writing, but perspicuity in proving, as will be seen hereafter. Note that wheresoever − is not expressed, there + is understood, though it be not expressed. Also in trigonometry. I use, s. pzs, for the sine of an angle pzs, and s.c. zp for the sine of the compliment of a side pz to 90. Also, t. zp and t.c. spz for tangent of zp and tangent of the compliment of spz, etc. Also for Radius I use r. If the sign of Addition, namely + stand before any quantity, it shows that quantity, to be more than nothing; that is something. But if the sign of Substraction, to wit − stand before any quantity; it shows that quantity to be less than nothing: or a want of the said quantity. So + 4, signifies four of any thing: but − 4, signifies a want of four, or four less than nothing. In ADDITION. The addition of a want of any thing, is all one with the subtraction of the same thing. So if to + 12 you add − 5 it makes + 7 And if to + 12 you add − 16 it makes − 4 But if to + 12 you add + 16 it makes + 28 In SUBTRACTION. The subtraction of − is all one with adding + So if from + 12 you subtract − 5 remain. is + 17. And if from + 12 you subtract − 16 remain. is + 28. Addition of + to + and Subtraction of − from − is all one with Common Addition and Subtraction. And generally for both. In Addition, add the quantities together with the same sign. In Subtraction, add them also, but all the signs of that which is to be subtracted from the other, must be changed. Example. If to + 6 − 2 + 3, be added + 5 + 1 − 3 the sum is + 6 − 2 + 3 + 5 + 1 − 3 = 10. But if from + 6 − 2 + 3, be subtracted + 5 + 1 − 3, the remain is + 6 − 2 + 3 − 5 − 1 + 3 = 4. This Rule is general, and generally known. In MULTIPLICATION. + multiplied by + ever produceth + + multiplied by − ever produceth − − multiplied by − ever produceth + More Varieties there are not. The quantities that are accompanied with these signs of + & − (in both Multiplyers being placed one under another, as in common multiplication) must be multiplied every one below into very one above, and then this work is done. So if, + bb + b − c, be multiplied by + f − g place them thus. + bb+ b − c + F − g Saying, + f multiplied into + bb gives + bbf And + f into + b gives + fb And + f into − c gives − fc And − g into + bb giveth − bbg And − g into + b gives − bg Lastly, − g into − c gives + cg Which added together is, bbf − bbg+ fb − fc − bg+ cg Which is the true product. In DIVISION. If the line of separation do not serve the turn that is, if any desire, (and it may be done) otherwise, it must then be by seeking what quantity may be multiplied by the Divisor to produce the Dividend. So if bb + bc − bf − bg − cg + fg, were to be divided by, b + c − f, trial must be made what mixed quantity multiplying b + c − f will produce bb + bc − bf − bg − cg + fg, In which there is this of Compendium, that seeing the Dividend consists of six members, and the Divisor of three, the quotient must be of two; that is a Binomial only. And because the quantity g is found in the Dividend, and not in the Divisor, it must be in the quotient. The said quotient therefore must be one of these, b + g, b − g, or g − b. It cannot be the first, for + g, into − f would have produced − fg: but in the Dividend it is + fg, therefore it must be − g. By the same reason it cannot be the last, as also because − b, into + b produceth − bb, but it is + bb, in the Dividend. The quotient sought, must therefore be b − g. Some further Rule for saving labour herein might be given: but every one likes that best which he finds out himself. Nor is it my purpose to write a Book of Algebra; but to premise so much of the Rudiments thereof, as the Reader may stand in need of in perusing the following Treatise. Wherein because Division is seldom needed; If I have a little exceeded already, and shall a little more in treating (but very briefly) of resolving some few Rooted Aequations, I shall ask the Readers pardon for both together. CHAP. II. Of Aequations. AN Aequation is when one or more special quantities, are equal to one or more other special quantities, and written with the sign of equality betwixt them; As aa = bb. This is called a simple Square Aequation. And bb, being a known square, the square root thereof being extracted, is equal to a. And that is the thing required. But, aa + ba = cc, and aa − ba = dd, and lastly − aa + ba = ff; are all of them of that kind, which are called mixed aequations, because a (the thing required) is multiplied not only into itself, but into another known quantity, namely into b. And note that this known quantity in all mixed aequations is called the Coefficient. Note also that the three sorts of mixed aequations above expressed are all that can happen in quadratiques: And by some one of these, all Problems whatsoever transcending plain Division, and not reaching Solids, are to be resolved by finding the Root a, according to these Old Rules. In the First, aa + ba = cc. Unto the quantity given namely cc, add the Square of half the Coefficient, it makes + cc+ bb/ 4 Which if it be in lines, may be reduced into one Square, and from the side of that Square, take half the Coefficient, and the remainder shall be a. Which was the thing desired. In the second, aa − ba = dd. Unto dd add bb/ 4 as in the former, and the sum thereof being always in numbers a Square, or in lines to be reduced to a Square as aforesaid: Unto the Root or side of that Square, add half the Coefficient, the Sum thereof shall be a, or the Root of the Aequation sought for. In the last, − aa + ba = ff. From the square of half the Coefficient, which is bb/ 4 take the quantity given, that is ff there will remain bb/ 4 − ff, which being put into one Square, and the side thereof known: If that side be either added to half the Coefficient, or subtracted therefrom, either the sum of that addition, or the remain of the subtraction, is equal to a. For all Quadratique Aequations of this kind (where aa the greatest unknown power is wanting) have two Roots, which being both together ever equal to the Coefficient, if upon the Coefficient, as a Diameter, a Semicircle be described, and the side of ff (the quantity given) be applied therein, perpendicular to the Diameter b, the two segments of b are the two Roots sought. For in the Aequation − aa + ba = ff, it is by the 14th. of the 6th. of Euclid, as followeth. mathematical diagram b − a′ f″ f′ a″. Wherefore either Segment may be a, and the other will be b − a, and f a mean betwixt them. Likewise in the two former Aequations, the work may be effected Geometrically, & proved also by this present Scheme. mathematical diagram In which, as the figure intimates, the Perpendicular represents the side of cc in the first Aequation, and the side of dd in the second. mathematical diagram Draw a line bg from the centre b to the top of the perpendicular, the centre b being first taken in the middle of the line b, to wit, of the Coefficient, for so it is usually called. And first, let the pricked line be put for a, Therefore, by the before recited Proposition, It is, a + b′ c″ c′ a″. Eucl. 6.14. And if (as the Rule prescribeth) to the square of half b you add the square of c, the total shall be the square of the line bg: By the 47th. of the first of Euclid, If therefore from the line bg, or (which is all one) bf, you take the line br, which is half the Coefficient b (for the whole Coefficient b, is the same with sir) the rest, namely the pricked line rf, shall be equal to a. For, rf = ns = a. In like sort concerning the second Aequation, aa − ba = dd: If according to the Rule, you add the squares dd, and bb/ 4 together, it gives the square of the line bg, to the Root of which, to wit, bg, if you add half the Coefficient, to wit, br, or bs, the sum shall be fs or nr, either equal to a. And then, as nr, that is a, is to d, so is d to rf, or a − b, as it ought to be. I intent anon to write something of Extraction of Roots, according to the general Method of resolving all manner of Aequations of Powers, how high or composed soever. I do not mean to exemplify them any further than the Cubique order. There are Authors enough, whom they that desire the full of that Artifice, may at their own leisure in Books confult. And now because I shall herein make some use of Aequations, though not higher than cubics, or at the most the Biquadratique order: I think fit to admonish the Reader, that in putting a always for the thing sought, and working therewith, as if it were known, quite through as the question requires, he shall at last come to an Aequation, but it may be such a one as wants reducing: of which a little. REDUCTION Of Aequations is done by adding all that's necessary, or subtracting all that's not necessary on both sides the sign of equality: Or by subtracting contradictories if they happen on one and the same side, until the Aequation, purged of all unnecessary members, remain with all that's absolutely known on one side, equal to (as little as may be) all that's unknown on the other side. One example of this shall serve as followeth: In the Aequation aa − ba + dc + ba = gg + ba − dc. To reduce this, you must remember what hath been said before; that the taking away a Want of any thing, is all one with the addition of that thing. Therefore seeing there is on the first side a Want of ba expressed by − ba, if you take away that − ba, you thereby add ba on that side. Wherefore that it may still be an Aequation, you must add ba on the other side also. Then it will be, aa + dc + ba = gg + 2 ba − dc Again, subtract ba on each side, than it is, aa + dc = gg + ba − dc Once more, subtract ba on each side, that you may bring it to that side where aa stands. Then it is, aa − ba + dc = gg − dc. Lastly (that the Consonants, or known things may come all on one side) subtract dc on each. Then it will be, aa − ba = gg − 2 dc. Take the Rectangle 2 dc out of the Square gg, and let the rest be a Square, namely ff. Then it is reduced, aa − ba = ff. Having gone a little about, only for exercise of them that are quite unskilful herein, now they shall see this Reduction might have been quickly done another way, that is, seeing in the Aequation aa − ba + dc + ba = gg + ba − dc There are in the first part Contradictories, to wit, − ba and + ba, they (destroying one another) might be taken away both at once, So it will be, aa + dc = gg + ba − dc. Then if you subtract dc and ba on both sides, it will be reduced to aa − ba = gg − 2 dc, as it was before. And gg − 2 dc being put into one Square ff, the Aequation aa − ba = ff, may be resolved as the aequation aa − ba = dd was, by the second Rule for plain Aequations, a little before expressed. And as here the Reduction was made by Addition and Subtraction only, so sometimes it is made by Multiplication, sometimes by Division; in both or either of which, this is general: that Whatsoever is done to any one Member, must be done to every Member quite through the Aequation. CHAP. III. Of the resolution of Aequations, according to the general Method composed by Mr. Tho. Harriot. ALthough (having before showed Rules for all sorts of mixed squares) it may seem preposterously done hereafter to speak of Simple Squares; yet forasmuch as I pretend not much to Method or Order, and because the general Method of Mr. Harriot gins with Squares, I will do so, but only with one Example. That is, Let there be an Aequation of aa = ff. Or let it be exhibited in numbers, aa = 69169 First, take notice that all Squares whether simple or mixed in Numbers, are to be marked with points, the first always over the place of Unity or unities, and so successively every Binari or second figure. Cubes with every ternary figure. Biquadratiques with every quaternary. Sursolids, every quinqnenary, and so forwards. This square number so pointed is 6̇91̇69̇ In which because there are three points, there are three figures in the Root. So that a being a single letter cannot fitly represent that Root, but some trinomiall, as is b + c + d should be put equal to a, and the square thereof should be equal to aa, or 69169. But because it may be done aswell by adding the Gnomon, that is repetition of the second working, (as they are commonly called) so often as the points are more than two; a binomiall will serve (with less trouble) to do the same. Let that Binomiall be b + c. And put b + c = a. Their Squares shall be therefore equal. That is, bb + 2 bc + cc = aa. That is, bb + 2 bc + cc = 69169. The Resolution. The homogeneal number given 6̇91̇69̇ First single Root b = 2 and bb = 4.0000 Which 4.0000 being subtracted from the number given 69169, then there Remains of the Number given 291̇69̇ Remains of the Number given, 291̇69̇ Root decuplate b = 20 Divisor 2. b 40.00 The second single root c = 6 2. bc 240.00 cc 36.00 276.00 Subtract 276.00 Remains of the Number given 15669̇ The Root increased b = 26 Root increased and decuplate b = 260 Divisor is 2 b = 520 The third single Root c = 3 2 bc 1560 cc 0009 Totall 1569 Subtract 1569 Remains of the Number given 0000 The Root increased 263, is therefore the true Root, as may be proved by recomposition, or multiplying 263 by 263, for the Product will be 69169, which was the number given. The cyphers which are put after in the Divisors and Subtracts, are only to fill up the number of places, by which the number given, or rather the remaining Points would else exceed. For the like purpose is used the decuplation of the Roots, as only to supply a place until another figure succeed in place of the Cipher. And in nothing else doth this work differ from the ordinary Extraction of the Square Root, commonly taught and known. The reason of it depends upon the 4th. Prop. of the second Book of Euclid, where it is demonstrated, that If a right line be divided by chance into two parts, the Square made of the whole, is equal to the Squares of the parts, and to the Rectangle made of the parts twice. So it is here as followeth. The Square of the greater part, that is, of 260 bb = 67600 The Square of the lesser part, that is, of 3. cc = 00009 The Rectangle of the parts, that is, 260 into 3 twice. 2 bc = 01560 Equal to the whole Square. 69169 Nor do these letters represent so naturally the things themselves in a divided Superficies only, but as properly and clearly the parts of Solid Bodies, of which, two or three Examples for satisfaction. In which I admonish the Reader, to be intent to the several pointings of the quantities according to their due order, as is before expressed, and also to the placing of the Divisors and Substracts by cyphers, as before also is intimated: for this to the Ingenious is enough, & a long verbosity to others will scarce be so. Of cubical Aequations. Let there be a Cube aaa = fff Or proposed in Numbers aaa = 41781923 Put (as before) b + c = a Then their Cubes also shall be equal. That is bbb + 3 bbc + 3 bcc + ccc = 41781923 The Resolution. The Homogeneal Number given 41̇781̇923̇ The first single Cubique Root b = 3 And bbb = 27.000000 Subtract 27.00000 Remains of the Number given 14781̇923̇ Remains of the Number given 14781̇923̇ The first Root decuplated b = 30 Divisor 2790.000 Second single Root c = 4 Subtract 12304.000 Remains of the Number given 02477923̇ The Root increased b = 34 Root increased decuplate b = 340 The third single Root c = 7 Subtract 2477923 Remains lastly of the number given 000 The Root increased b + c = 347 Which is the true Root of the Cube 41781923, as may be proved by recomposition, that is, by Multiplying 347 by 347, and the Product again by 347, the last Product shall be equal to the Cube which was given to be resolved. And as above in the Square the Canon of the Resolutions was the letters bb + 2 bc + cc, being the true Square of b + c. And those letters did answer exactly to the parts of the Square divided alike in both the Dimensions: So here also the Canon of Resolution, or the letters bbb + 3 bbc + 3 bcc + ccc, do exactly answer to the Parts or Members of a Cube, divided into two parts, alike in all the three Dimensions, as any one may prove upon a Cube made of some slender matter, and cut through all three ways, for he shall find the whole Cube (supposed equal to 41781923 as before) justly made up of the two Cubes of the two segments, that is, bbb and ccc, and three Parallelepipedons, whose length and breadth are equal to b, and their thickness to c, those three are the 3 bbc. And lastly, three other Parallelepipedons, whose length and breadth are equal to c, and their thickness to b, such are the 3 bcc. See the following Schematisme. The Cuhe of the greater Segment which is 340, bbb 39304000 The three greater Parallelepipedons, 3 bbc ·2427600 The three lesser Parallelepipedons, or 3 bcc ···46980 The Cube of the lesser Segment, which is 7, ccc ·····343 The whole Cube given 41781923 Note, That the greater Segment is the aggregate of all the single Roots except the last, being duly valued by a Cipher, as here it is 340, but the lesser segment is the last single Root only, as here 7, I have done this to let the Reader see, that he may be sure let the quantity to be resolved be great or little whatsoever, if he be careful to make his Canon right, the letters themselves will direct him how to frame his Divisors and Subtracts in order to the final resolution, especially in these unmixed Quantities, where the points limit how far the subtract shall advance at every operation, beginning first at the point next the left hand, not further, and to the second point only at the second work, and not otherwise in all that follow. And in Mixed Aequations, if they be made up of Cube with addition of certain Squares, or certain Roots, or both Squares and Roots, or by Subtraction of the same, the Canon of the Resolution must ever be made by multiplying the assumed Root b + c in the place of the quesititious Root a, quite through the Aequation in all the degrees thereof, for so shall arise all the several parcels of which the several Subtracts are orderly to be made. In a Cubique Aequation, if all the quantities be present, there is no need to point any but the cubics and Roots: yet I have here distinguished the places of the Squares also with little Crosses obliquely; which labour, when the Workman is intent upon his business, may well enough be spared. Of the resolution of Mixed cubics. Let the Aequation aaa + daa − ffa = ggg be proposed in Numbers. As let it be aaa + 32 aa − 75 a = 29282970 Therefore d = 32 and ff = 75 And ggg = 29282970 Put b + c = a And make the Canon of Resolution by substituting b + c in the place of a quite through the several quantities aaa + daa − ffa. The Canon rightly made will be + bbb + 3 bbc + dbb − ffb + 3 bcc + 2 dbc − ffc + ccc + dcc These several parcels of the Canon, being rightly subtracted from the homogeneal Number 29282970, the Number shall be thereby resolved, and the Root a found. ☞ Note first, That all the parcels in the Canon, which have not the secondary Root c in them, as + bbb + dbb and − ffb, are to be subtracted at the first Operation, the other remaining parcels to be all subtracted as often as there shall be points left above. The Resolution. The homogeneal Number given The first single Root b = 2 Subtract 92650.00 Remains of the Number given Remains of the Number given The first Root decuplate b = 20 Divisor 1390450 The second single Root c = 9 Subtract 17793450 Remains of the Number given The Root increased b = 29 Root increased decuplate b = 290 Remains of the Number given Divisor 271655 Third single Root c = 8 Subtract lastly 2224520 Remains of the Number given 000 Whereby it appears that the whole Root 298 is the true Root whereby this Aequation is explicable, as may be proved also by recomposition. For bbb = 24389000 3 bbc = .2018400 3 bcc = ... 55680 ccc = ..... 512 dbb = .2691200 2 dbc = ..148480 dcc = .... 2048 In all = 29305320 Fron which subtract ffb + ffc = ... 22350 Remains 29282970 Which was the whole Homogeneal Number given. NOTE. Whereas in composing the Divisor all the gradual quantities are used, as in the former example, 3 b and d, aswell as 3 bb and 2 db, it is to be noted that in practice, those smaller particles 3 b, etc. May be omitted; the other without them ministering light enough for choosing the Secondary Roots. Having now instanced in an Example where all the powers were present, in these one or two that follow, to make the work shorter, I shall leave out one or other of them. In the Aequation aaa + ffa = ggg. Propounded in Numbers aaa + 320406 a = 8348132, It sometimes happens that the Coefficient abounds with more binary figures then the homogeneal doth with ternaries, in such a case that there may be rooom made to begin the Extraction. The Coefficient must be devolved to the next point further to the right hand, or to the second, third, fourth, or further, if need require, and there the work is to begin. The Coefficient is always the known quantity which multiplies any of the unknown inferior quantities. Example of Devolution. aaa + 320406. a = 8348132 Put b + c = a The Canon will be bbb + 3 bbc + 3 bcc + ccc + ffb + ffc = 8348132 Resolution. The Homogeneal number given The first single Root b = 2 Subtract 641612.0 Remains of the Number given The first Root decuplate b = 20 Divisor 321606 The second single Root c = 6 Remains of the number given Subtract 1932012 Remains of the number given 000 Wherefore the whole Root is equal to the Root increased, 26, as may be proved in manner as beforesaid. It sometimes happens also in the Aequation aaa − ffa = ggg Put into Numbers. As aaa − 105000. a = 203125. That the Coefficient abounds with more binary figures then the homogeneal with ternaries: Wherefore that there may be place for the Resolution, put before the homogeneal, toward the left hand, so many cyphers as will afford that to receive as many cubical points, as the Coefficient doth Quadraticall: And at the first empty point, as it were by anticipation, begin the Resolution. In which there is this of Compendium, that the first Square Root extracted out of the Coefficient, is either equal to the first single Root of the homogeneal sought, or less than it by Unity. But if the Aequation had but two Dimensions, As aa − 254 a = 65024, than the first figure of the Coefficient, namely 2, is the first Root. Example of Anticipation. The homogeneal Number given The Canon is b + c = a bbb + 3 bbc + 3 bcc + ccc − ffb − ffc The Resolution. The first single Root b = 3 Subtract the difference which is − 45000.00 Remains of the Number given The first Root decuplate b = 30 Divisor 165000.0 Subtract Remains of the Number given Remains of the Number given +1035125 The Root increased b = 32 Root increased decuplate b = 320 Divisor The third single Root c = 5 Subtract +366800.0 Remains of the number given Which showeth that the Root increased, b + c = 325, is the true Root of the Aequation, And it may be proved by recomposition as formerly. In the Aequation − aaa + ffa = ggg, Which is explicable by two Roots, as shall be showed in the next Chapter, Section 5, to find them both. Put the Aequation into Numbers. As − aaa + 52416 a = 1244160 Therefore ff = 52416 & 1244160 = ggg Put b + c = a Therefore = 1244160 Extraction of the greater Root. The Homogeneal Number given 1244160 The first single Root b = 2 Subtract Remains of the Number given The first Root decuplate b = 20 Divisor The second single Root c = 1 Remains of the number given − 1239040 Subtract − 736840 Remains of the number given The Root increased and decupled b = 210 Divisor − 79884 The third single Root c = 6 Subtract − 502200 Remains of the number given 000 Root increased b + c = 216, which is the true Root sought. 2. Eduction of the lesser Root by Devolution. The homogeneal Number given Subtract +1040320 Remains of the Number given The Root increased and decupled b = 20 Divisor .51216 The second single Root c = 4 Subtract +203840 Remains of the Number given 000 The Root increased b + c = 24 Wherefore 24 is the true Root sought, as may be proved by recomposition, as hath been showed before. So this Aequation is explicable by two Roots, that is, 216, and 24. VIETA, Lib. de Recognitione aeqnationum, Cap. 18. Prop. 2. saith, That in the Aequation − aaa + ffa = ggg, the Coefficient ff is composed of three proportional Squares, and the Homogeneal ggg is made by Multiplication of the aggregate of the two first, or the two last, (for all is one) into the side of the other, and the Root a may be the side either of the first or third. This (or the same in substance) saith that Noble Author, And it is evident, for make cc′ + dd″ + hh′″ = ff And pnt c = a Therefore ccc + ddc + hhc − ccc = ddc + hhc Or put h = a It is hhh + ddh + cch − hhh = ddh + cch Both which are manifest COMPENDIUM 1. Hence it may be showed, that either of the quaesititious Roots, as a, being found and called c, the other Root e may be found by a Quadratique Aequation only. For supposing ee + ce = ff − cc, Than It is ee + ce + cc = ff. And cc′ ce″ ee′″ Euclid 6.23. But by construction cc′ dd″ hh′″. And cc + dd + hh = ff. So then hh = ee and h = e. But it was showed before that h might be a Root of this Aequation − aaa + ffa = ggg And therefore e also is a Root of the same, and the Compendium is proved. Example also in Numbers. In the last Aequation aa = cc = 46656 And ff = 52416 From which take cc = 46656 Remains ff − cc = 5760 05760 But And The Sum is ee + ce = 5760 Therefore ee + ce = ff − cc, Which was, etc. In the Aequation − aaa + faa = ggg, the Coefficient f is composed of three proportional lines, and ggg is equal to a Solid made by a Square (whose side is equal to the two first, or the two last) multiplied into the remaining line: And the aggregate of the first and second may be a, and the aggregate of the second and third shall be e. Put 1′ 2″ 4′″ And suppose − aaa + 7 aa = 36 Then a may be 3, and e is 6. Vieta, de Recognit. Cap. 18. Prop. 6. COMPENDIUM 2. And therefore the Root a found, and called c, the Root e may be found by a plain Aequation; for suppose the middle proportional y, it is f − y − c′ y″ c′″. And fc − cy − cc = yy Or, yy + cy = fc − cc. And making fc − cc = xx, it is yy + cy = xx. And the Root y being found by the first Rule of Chap. 2, It is lastly (making c′ y″ d′″) y + d = e. I will here add a few Rules (grounded upon Mr. Harriots 6 Sections) by which the Reader may easily perceive the Fabric of Aequations, their Roots, increment and decrement, Multiplication and Division of them, and their number in any Aequation as followeth. CHAP. IU. Rule 1. EVery Aequation being composed of some known and some unknown quantities hath its Original by roots eomposed of a quantity known and of one other quantity unknown, and these roots multiplied together produce certain particular Members with + and − respectively signed (for in every aequation both these signs are present) which orderly placed make up the aequation. As the aequation aa − ba − ca + bc = 0. is made by multiplying a − b = 0 by a − c = 0. And because it was at first a − b = 0 therefore a = b and the like of c. And from hence it follows that where the first term (or highest power) in a quadratique Aequation is signed − there the Aequation hath two roots, as here by substracting on both parts + aa − ba − ca, the Aequation will be bc = − aa + ba + ca, and must have 2 roots. 1 These compound quantities so multiplying I shall call Binomialls, whether a + b or a − b. not having any need in this treatise to distinguish betwixt Binomials and Residuals. 2 The aequation aa − ba + ca = bc. If it be, b < c. put c − b = d. than the aequation will be, + aa + da = bc, and is of the first kind mentioned in Chap. 2. but if it be b > c, put b − c = f and the aequation will be + aa − fa = bc, and is like the second sort in the same Chapter. The Original of the aequation aa − ba + ca − bc = 0 here proposed, is + a − b = 0 multiplied by + a + c = 0, that is a = b by a = − c. This aequation hath but one true root, which is b, and one false, which is c. 3 By this which hath been said it is plain that some aequations have as many roots as dimensions, some not so many, but none can have more; for the number of dimensions being the same with the number of multiplyers (if all divers) can be but all roots. Nor can the aequation be divided by any other thing than one of those Binomials by whose multiplication it was made. But if the multiplyers how many soever be still the same, there can be but one root. For let + a − b = 0 be muliplied Biquadratically, the product is + aaaa − 4 baaa + 6 bbaa − 4 bbba + bbbb. where it is plain there can be no other root but b. I mean none greater or less than it: because in truth here are 4 roots, but every one singularly equal to b. For if there may, let it be d, and let d be greater or lefle then b, it imports not which. And seeing d = a, Substitute d in the place of a, quite through the Aequation, it will be dddd − 4 bddd + 6 bbdd − 4 bbbd + bbbb = 0. Which if d > b, or else d < b, is at first sight impossible: For the difference between the + and − is always equal to the power of the difference between b and d, which power is here a Biquadrat, therefore d = b. And again seeing this Aequation may be derived by putting b equal to d, for substituting b in the place of d quite through, It will be + 2 bbbb + 6 bbbb = 4 bbbb + 4 bbbb Which is manifest, therefore again b = d, which is contrary to the supposition, therefore b is the only Root of this aequation, for indeed, the aequation proposed being made only of multiplications of a − b = 0 cannot be divided, that is resolved, by any other Binomiall than a − b, of which it was made, 4 Hence it is that the last term in every aequation may be called the Homogeneal, because it is naturally made by multiplication of the Roots of the aequation, though the coefficients in some ordinary aequations are disguised with other characters, which happens by Addition or Subtraction of them, to reduce the canonical aequation to fewer members, whereby the redundancy of the signs + and − is to be taken away, this is to be seen above in this Rule, where the aequation + aa − ba + ca − bc = 0 is reduced to + aa + da − bc = 0, by making d = c − b and + aa + ba + ca − bc = 0, reduced to + aa − fa − bc = 0, by making b − c = f Where the Coefficient d or f, is not a part of the Homogeneal bc, but a difference by which b is greater or less than c: by help of which difference, the aequation which consisted canonically of four Members, hath now but three. 5 And this Reduction is useful, for as Mr. Des Cartes saith, and which may be seen true by the way of Multiplication above showed, every aequation hath so many true Roots as the Signs + and − therein are changed, which in the canonical aequation + aa − ba + ca − bc = 0, are changed three times, whereas the aequation hath not three true Roots, but one true and one false, that is b & c, and the common aequation reduced changeth the signs but once, that is, from + da to − bc in the former; or from + aa to − fa in the later: and from thence it may be known that the aequation hath but one true Root. The like consideration ought to be in others. And whereas the said Des Cartes doth often mention false Roots, it is to be noted that such are less than nothing, as + a + b = 0: Or + a = − b, & if any true root, as + a − c = 0 be multiplied by this + a + b = 0, there will arise an aequation + aa + ba − ca − bc = 0 where the sign + follows twice, the sign − twice, and they are once changed, which should intimate (according to Des Cartes) two false Roots, and one true: for he saith, So many times as + or − come twice together, so many false roots there are, this aequation therefore must be reduced, by making b − c = d if b > c, or else if b < c than make c − b = f, so it will be either + aa + da − bc = 0, Or + aa − fa − bc = 0 which confirms that which Des Cartes saith of twice + or −: Namely, that there are as many false roots in the aequation, as + or − come twice together, and so many true roots as + and − are changed. And where the Roots are all false, the aequation is impossible, as a + b = 0 multiplied by a + c = 0, produceth aa + ba + ca + bc = 0 which cannot be. And therefore when there is an aequation pretended like aa + ba + ca = − bc, present judgement may be made. 6 The same Des Cartes saith also that all the false Roots in any aequation, may be turned to true ones, and the true ones to false, by changing the signs of the second, fourth, and every even term. And this is evident, for of the aequation a4 − 2 a3 + 10 aa − 30 a − 87 = 0 by such change is made + a4 + 2 a3 + 10 aa + 30 a − 87 = 0 where the first had three true Roots, and but one false, the later hath three false and but one true. This Aequation was taken at all adventures, to serve for an Example only, whereas any other whatsoever will do the like. Rule 2. The unknown roots of an Aequation may be increased or decreased, by supposing another unknown quantity + or − the decrement or increment, and of that Binomiall composing the aequation as it was before of the first unknown quantity: and if this increment be put equal to such a part of the coefficient of the second term, as unity is of the dimensions of the first term (if the signs of the first and second be both + or both −) or if the Decrement be made equal to such a part of the said coefficient as unity is of the dimensions as aforesaid, (if the signs of the first and second term be one + the other −) then by such increase or decrease of the root the second term of the aequation shall be taken away, and awlled. Example. In the aequation + aaa + baa − bbc = 0, the Root a may be increased by making e − q = a, and substituting e − q in the place of a quite through the aequation, and thereby shall arise a new aequation: Which is equal to the former as you see agreeing in the particulars, and the root e being found, a may be had by casting away q from e. And because the number of the dimensions of the first term aaa is 3, if according to the later part of the rule the quantity q be proportioned, by making 3′ 1″ b′ q″ then b = 3 q and + be will destroy − 3 qee, and so the second term ee will be quite taken out of the aequation, as is manifest, for the aequation so purged will be + eee − 3 qqe + 2 qqq − bbc = 0 And by subtracting on each patt + 2 qqq − bbc having first made bbc − 2 qqq = ddd, it will be then + eee − 3 qqe = ddd. The manner of such reduction of Solids, shall follow in the next Chapter. In like sort the Root a, might have been decreased by any quantity, as x, which if it be proportioned to b as aforesaid, would take away the second term of an aequation, where the signs of the first and second terms are not like; as in the aequation + aaa − baa − bbe = 0, by putting 3 x = b, and e + x = a: the Problem will be fully performed by making e + x the Root of the new aequation, as before was e − q, observing the same order in composing the particulars, due respect had to the signs + and −, where they ought to be altered. The former reduced aequation + e3 − 3 qqe = ddd might be further reduced (if need require) to + e3 − bqe = ddd. NOTE. This augmentation and diminution of the Roots in such manner as to take away the second term of any aequation, is of excellent use in such aequations as have three or four dimensions, and cannot by any division with any binomiall made of a + or − some other known quantity, as b, c, or the like, be reduced to fewer dimensions, whereby it is certain that such an aequation is Solid, and cannot by any artifice already, or likely to be invented, be resolved by Ruler and Compass, but by any of the Conique Sections it may; in this case it is either necessary or extremely facilitating, to take away the second term (if there be any) from the aequation, as shall be seen hereafter in its place. Rule 3. The unknown Root of any Aequation may be multiplied (or divided) by any known quantity multiplying (or dividing) the second term of the Aequation by the said quantity, the third by the Square, the fourth by the Cube thereof, and so forward continually in this order, as often as there are terms in it, having first assumed another unknown quantity, so multiplex to the said unknown Root, as is required. Example. In the cubical Aequation a3 + baa + cca − bcd = 0: Let it be required to multiply the Root a by 4. Assume e = 4 a and write eee + 4 be + 16 cce − 64 bcd = 0 Which is an aequation, and the root e is quadruple to a, as may be proved thus. Put a = 4 b = 3 c = 2 and d = 21⅓ Then Therefore a3 + baa + cca − bcd = 0 Again, Put e = 16 All else the same still. Then Therefore e3 + 4 be + 16 cce − 64 bcd = 0 And e = 16 = 4 a Which was to be proved. The utility of this Rule will appear in reducing aequations affected with fractions, to whole numbers, by multiplying the Roots by the denominator or denominators of the fraction, for by such means the coefficient of the second term is multiplied by the same as before, multiplying a by 4, multiplied also b by the same number 4. And many times by this Rule aequations may be freed from Surd Numbers also: especially if such be found in the second term, as is easy to be seen by trial, for if there be an aequation so affected, As aaa + √ 8 aa + 29/24 a − 4 √ 2 = 0 Put e − √ 8 = a And write + eee + 8 ee + 9⅔ e − 128 = 0 So the Surds are vanished. But if yet it be required to avoid the Fraction 9⅔ e, then make y = 3 e. And multiplying 8 by 3, 9⅔ by 9, and 128 by 27, there will be a new third aequation. + yyy + 24 yy + 87 y − 3456 = 0 Which consists of entire Numbers, having one true Root which is 9, and the Root of the middle aequation was 3, which is the third thereof, and the Root of the first aequation was 3 − √ 8 And now I hope this Rule and the use of it is plain enough. NOTE 1. It may be noted, that if the Surds in the second and last terms of the first aequation, to wit, aaa + √ 8 aa + 29/24 a = 4 √ 2 had been utterly incommensurable, the reduction had not been so fesible. For although 4 √ 2 multiplied by the cube of √ 8 that is by 8 √ 8 produceth 32 √ 16. which is equal to the entire number 128, yet if it had been 2 √ 3 or 2 √ 5, or any such primes to be multiplied by 8 √ 8 the product would have been 16 √ 24 or 16 √ 40. though this last may (by the note after the consectary in Chap. 6.) be reduced by multiplying it again by √ 40 unto the entire number 640. Nevertheless this second multiplication by a Surd, renders the aequation inexplicable, at least by the precedent Rule. NOTE 2. It may be further noted, that if instead of e = 4 a one would put e = fa lines not being so liquid as numbers, the aequation would then be eee − fbee + ffcce − fffbcd = 0 increasing the dimensions of the lesser terms, for remedy whereof three lines are to be found in proportion one to another as are the magnitudes fb. ffc. fffb. of which let the first line be supposed to contain Unity as often as the superficics fb doth (for which purpose Unity must be a line set, and agreed on before.) The names of these lines when found may be called g, h, k. and the aequation may be written + eee + gee + hce − kbc = 0. NOTE. 3 But it is again to be noted, that where the lines f, b, and c, are commensurable in length the three lines k, h, g, may be very easily found, for than they may be signified by numbers and if f be put for Unity then e = a and the work frustrate, but where the said lines are incommensurable in length this Reduction is always hard if not impossible: For those incommensurable lines do most commonly represent such furred numbers as cannot by any Reduction be compared. Rule 4. The Aequation aaa − 3 bba = 2 ccc, or any other like it, by putting ee + bb/ e = a may if c > b be brought to eee = ccc + ddd or if c = b to eee = ccc, or lastly, if c < b then to eee = ccc + √ − dddddd: Which last may be called an impossible Aequation. Put e′ b″ bb‴/ e And because a is equal to the sum of the Extremes, which are e + bb/ e therefore, From thence it will be Therefore rejecting the contradictories, and multiplying all by eee, it is, + e6 + b6 = 2 c3 e3. Therefore + e6 − 2 ccce3 = − bbbbbb. And + e6 − 2 c3 e3 + c6 = + c6 − b6. Therefore, (for e3 − c3 = √ e6 − 2 e3 c3 + c6) + e3 = ccc + If now in the first case c be greater than b, then put c6 − b6 = d6. Then it will be eee = ccc + √ dddddd That is, eee = ccc + ddd. Which is the Aequation promised in the first case. Secondly, If b be equal to c, then c6 − b6 = 0 And it will easily follow, seeing (as is showed above) that e6 − 2 c3 e3 + c6 = o, therefore the Root of it e3 − c3 = o, that is eee − ccc the second aequation prescribed. Lastly, by the third case, seeing c is less than b, Put cccccc − bbbbbb = − dddddd: Then it will be eee = ccc + √ − dddddd the equation prescribed in the third case, and (because of the inexplicability of √ − dddddd) impossible. COMPENDIUN Whereas Mr. Harriot saith Propter √ − d6 inexplicabilitem, &c. The said quantity √ − d6 is not explicable because − d6 ariseth d³ multiplying + d3 by − d3 betwixt which two there is no mean; for no one thing can produce d6 but d3 only, and − d6 is not produced by + d3 or − d3 because by both, this therefore may serve for a Compendium to save labour which might else be lost, in seeking that which is impossible to be found. NOTE. I use b6 for bbbbbb, and b4 for bbbb, and b3 c3 for bbbccc, and the like, (as Des Cartes hath done) only for abridgement, as in the Definitions of the Powers is already showed. And with that line over to distinguish betwixt √ c6 − b6 as one quantity, and √ c6 taken by itself and − b6 taken apart also, for by such mistakes may great errors succeed. I will add no more rules, these 4 may be multiplied by any one that doth not find these sufficient for his purpose, at his own pleasure. CHAP. V Of Reduction of Solids. HAving spoken in Chap. 4. Rule 2. of making bbc − 2 qqq = c⁶, and in Rule 4. of c6 − b6 = dddddd, I think it not amiss here to show how such Addition and Subtraction of Solids may be performed. And it may be noted that ddd, is for brevity sake there usurped for ggc. or some other solinomial rectangle Parallelepipedon. equal to the Binomial rectangle Solid bbc − 2 qqq, for if this Binomial could (by plain Geometry be) given in a Cube, as is ddd, something else might be done which here I will not speak of. Now therefore seeing b = 3 q as there it is, the aequation may be written 9 qqc − 2 qqq = ddd, or rather 9 qqc − 2 qqq = ggc. Make qq/ c = f, therefore 2 qqq = 2 qfc, Secondly, make 9 qq − 2 qf = gg, from thence it is plain that bbc − 2 qqq = ggc, which was first to be done. Secondly, to reduce c6 − b6 into one entire Solid, though not into a Squared Cube as d6, as is usurped by Mr. Harriot for brevity in writing, or facility in reasoning, Pag. 100, supposing that done which cannot be done by straight lines and circles hitherto. Now therefore seeing c6 − b6 is produced by multiplication of ccc + bbb into ccc − bbb. Make cc/ b = f, and bb/ c = g, & f + g = q and f − g = p, therefore bcf = ccc, and bcg = bbb, and bcq = ccc + bbb. Secondly also bcp = ccc − bbb: And therefore bbccpq = cccccc − bbbbbb, which was secondly to be done. Example in Numbers. Put b = 2 and c = 3: Then ccc3 = 729, and bbbbbb = 64, and then cccccc − bbbbbb, that is 729 − 64 = 665, which is produced by multiplying 27 + 8 by 27 − 8, that is, 35 by 19 Now make f = 9/2, and g = 4/3;, then f + g = 5⅚ = q, and f − g = = 3⅙ = p. And bcq = 35, and bcp = 19 And lastly bbccpq = 665. Moreover, if you make pq = xx, the Solid is further reduced to bbccxx, which although it be not a Squared Cube, yet it hath a square root, namely bcx, which may be of good use in many cases to resolve Aequations into Analogismes, of which kind of Demonstration, by help of Euclid 6.14. some notice is taken before in Chap. 2. NOTE. The three cases of the aequation a3 − 3 bba = 2 c3, mentioned in the beginning of the fourth Rule of Mr. last Chap. are called by Mr. Harriot, the first Hyperbolical, the second parabolical, the third Ellipticall, because of some similitude between them and those sections, of which three Cases, the first is resoluble by a Conique Section, the second by a Circle, and the third not at all. Multiplication and Division of Solids is altogether as easy as Addition or Subtraction, for if one would divide ccc by bb, make cc/ b = x, and again make cx/ b = z, then z is the Quotient required. Example in Numbers. Put b = 2 and c = 3, than ccc/ bb = 6¾, to find which, make cc/ b = x = 9/2, than cx = 27/2, & cx/ b = 27/4 = z = ccc/ bb = 6¾, c⁵ it should be. Again, if c5 should be divided by b, it is now ccc/ bb = z, and multiplying by b it is ccc/ b = bz: Again, multiplying by cc it is c5/ b = bccz, and bccz is the Quotient required. But if it be required to bring the quotient to a Biquadrat, make bz = dd, then ccdd = bccz And make cd = ff, than the quotient will be ffff. Multiplication is naturally so easy that there needs no more be said of it, than what hath been said already in Chap. 1. Now, of aequations consisting of 3 terms in continual proportion as a4 + bbaa = c4 or secondly a6 − bbba3 = c6, or lastly let it be − a8 + bbbba4 = c8, let them first be proposed in numbers as a4 + 2 aa = 24, if by Rule 1 of Chap. 2. it be wrought, it will be found √ 25 − 1 = aa, and aa = 4 or a = 2. Otherwise if the square of half the coefficient be added on both parts, than a4 + bbaa + 1 = 25. And their square roots also are equal; that is aa + 1 = 5 and aa = 4 or a = 2 as before. and the latter may prove the former. 2 a⁶ the second, let it be a6 − 10 aaa = 459 Add 25 to each part, than it it is aaaaaa − 10 aaa + 25 = 484. Now each part of the aequation is a Square & their Roots also are equal; that is aaa − 5 = = 22, that is aaa = 27, and a = 3. 3 Lastly, If − a8 + 700 a4 = 46875 from the Square of 700/2, that is, from 122500: take the homogeneal 46875, there remains 75625, whose square root is 275. And either 350 + 275. Or 350 − 275, that is either 625 or 75 is equal to aaaa, and a = 5. Or √ qq. 75 = a, which Charrcter √ qq. signifies the Biquadraticall Root. NOTE The first and last of these three aequations, may be done aswell in Lines as Numbers (by the said three Rules of Chap. 2. and so any aequation of 4, 8, 16, or. 32 dimensions, but aequations of 6, 12, or 24 dimensions, cannot be effected so, because there is ever one or more Cubique roots to be extracted, which without two means cannot be done. For if it may, than I say, that two means between any two lines may thereby be found, for in the second aequation a6 − bbbaaa = c6 by Rule 2 Chap. 2 c6 + ¼ b6 is a square, make cc/ b = d, then bd = cc, and bbdd = c4, and bbccdd = cccccc, then make ¼ bb/ c = f therefore fc = ¼ bb, and fcb4 = ¼ b6. Now because bb/ c = 4 f, make h = 4 f, then fccchh = ¼ b6. Make fc = ll, then cchhll = ¼ b6. Again, make bb + hh = mm, and dd + ll = nn. And then it will be .. ccmmnn = bbccdd + cchhll, that is, c6 + ¼ b6, to the square root hereof cmn, add ½ bbb: thus, make ½ bb/ m = p; then mp = ½ bb, and bmp = ½ bbb. Make bp/ n = q, than mnq = ½ bbb. Lastly, make c + q = x, than it is cmn + ½ bbb = mnx = aaa, by Chap. 2. Rule. 2. Now if m, n, and x be proportional, than the middlemost is equal to a, but that is uncertain, and cannot be made otherwise: But by making rr = mn it will be rrx = a3 and a will then be the lesser of two means between r and x if r < x or the greater means, if r > x. And so if r and x had been given, and required to find 2 means between them by retrogradation orderly, one might come to the said aequation a6 − bbbaaa = c6 of which if the root a be found, two means are also found between r and x which was to be proved. CHAP. VI Of Surd Numbers Rule 1 THe square root of any number being multiplied by that number, produceth the square root of the Cube of the number. For √ a multiplied by a produceth a √ a, but a √ a = √ aaa for taking Equimultiplices they will be equal, as if the first, namely a √ a be muliplyed still by √ a, the product is a √ aa, that is aa. And if √ aaa be multiplied by √ a it produceth √ aaaa that is aa also, wherefore a √ a = √ aaa. And therefore 3 √ 3 = √ 27 either of which is the cube of √ 3, and the like of all others. Rule 2. Surd numbers are multiplied and divided like whole numbers, the Product retaining still the Character of the Root .. That is, √ 2 multiplied by √ 3, produceth √ 6, and so of all others. NOTE. Where I shall have occasion (if any be) to speak of a Cubique Root, I shall sign it thus, √ c. and the Biquadratique Root thus √ qq. Rule 3. To multiply, divide, add or Subtract the roots of Surd numbers. And first of MULTIPLICATION. Besides that which hath been said in the last Rule above, these roots of Suids' may be multiplied and divided, and known by other names, so as sometimes the products, or quotient shall be rational. First, therefore any square root doubled is the square root of the quadruple, as 2 √ 5 = √ 20 and 2 √ 20 = √ 80. 3 √ 5 = 45. 4 √ 5 = √ 80. 5 √ 5 = √ 125. 2 √ 10 = √ 40. 3 √ 10 = √ 90. 4 √ 10 = √ 160, and 5 √ 10 = √ 250, &c, infinitely still multiplying the Numerator, 2, 3. 4, 5, etc. into itself, and the product into the snrd number, as if 3 √ 10 = √ 90, it ariseth from 3 times 3 into the furred number √ 10: and the like of all others whatsoever. For put √ a = √ 10, to be multiplied by another number, as by a = 10, the product is a √ a = 10 √ 10, which by the first rule is √ aaa = √ 1000, that is, the numerator 10 into itself making 100, which multiplied again by the furred √ 10, giveth √ 1000 And if it had been at first √ a = √ 10, multiplied by any other number, as e = 3, the product must by the same method be e √ a = e √ 10 that is (by the same reason as the former) √ eea = √ ee10 = √ 90. And it is plain, that if any Root be multiplied by 2 The product shall be the Root of the Quadruple. 3 The product shall be the Root of the Noncuple. 4 The product shall be the Root of the Sedecuple. 5 The product shall be the Root of the Vigintiquintuple. 6 The product shall be the Root of the Ttigintisextuple. And so forward infinitely, according to the proportion of the Squares of the Multiplyers Also by Decuplation, as if 5 √ 5 = √ 125, than 5 √ 50 = √ 1250. Or if 4 √ 4 = √ 64 then 4 √ 40 = √ 640. And (as above) if 4 √ 10 = √ 160, than 4 √ 100 = √ 1600. Also by Subdecuplation, if 2 √ 10 = √ 40, than 2 √ 1 = √ 4. Or if 5 √ 20 = √ 500, than 5 √ 2 = √ 50. And (according to that aforesaid) 3 √ 37 = √ 333, and 3 √ 36 = √ 324, that is, the square root of 3 times 3 times 36. And this may often be of use, not only in numbers but Species, and is therefore to be had in memory by him that would be ready in Multiplication of Surd numbers, or Surd quantities. Furthermore it may be useful to remember that in Reciprocal Surds as 4 √ 5 and 5 √ 4 these two have that proportion one to another as 4 hath to a mean betwixt 4 and 5. As for example 4 √ 9 hath that proportion to 9 √ 4 as hath 4 to 6, which is a mean betwixt 4 and 9, for 4 √ 9 = 12, and 9 √ 4 = 18, but 4′ 6″ 12′ 18″ or more generally a √ e′e √ a″ a′ √ ae″ for multiply the Means, it is ae √ a and multiply the Extremes it is a √ aee, & divide each of them by a the first is e √ a the other is √ aee, but by the former part of this rule e √ a = √ aee wherefore this is proved, CONSECTARY. Hence it is evident that roots of themselves inexplicable may be so multiplied as the product may be rational: for if √ 20, be multiplied by 4 √ 5 the product will be 4 √ 100 = 40. For 2 √ 5 = √ 20 and 2 √ 20 = √ 80, therefore 4 √ 5 = √ 80, but √ 80 multiplied by √ 20 giveth √ 1600 = 40, I need say nothing of Division, for that is no more but by the same steps to go back again, as √ 1600 divided by √ 80 quotient is √ 20. And so of the rest which hath been said in multiplication. NOTE. These things being so, it will not be hard to find some number to compare with any Surd number so as to make that work rational and exprimible which seemed not so: for there is not any furred number can be given which may not by some multiplication be made a rational number: for let it be √ 5, √ 7, √ 8, or any of these as √ 7 multiply it first by √ 7 that produceth 7, but multiply √ 7 by any square number whatsoever, as by 4 omitting the sign √, it gives 28, then again multiply √ 7 by √ 28 it produceth √ 196 = 14. For this is all one as to multiply one square number by another, which must needs produce a square number. So here the square number 4 was multiplied by 7 and after by 7, that is by 49, which multiplyers cannot produce any other than a square number, to wit 196 Euclid. 9.1. And whatsoever hath hitherto been said of Quadratiques, may serve for cubics also; due respect always had to the degree of the quantity and root, for any √ c. multiplied by 2 giveth 8 √ c. by 3 it gives 27 √ c. by 4 it gives 64 √ c. that is 2 √ c. 8 = √ c. 64 and 3 √ c. 8 = √ c. 216, and 3 √ c. 27 = √ c. 729 the proportion still increasing as the Cubes of their nultipliers. And the like consideration had, this may be applicable to Biquadratiques, or any higher order. And still whatsoever hath been said of multiplication, serves in a retrograde way for division also. Rule. 4 For ADDITION. Surd roots are usually added and substaacted by the signs + & − as the square root of 2 added to the square root of 8, sum is √ 2 + √ 8 or substracted rest is √ 8 − √ 2. But these may be added into one sum for seeing 8 is quadruple to 2 therefore 2 √ 2 = √ 8. And the sum is 3 √ 2 and the remain is √ 2. Likewise the reciprocal Surds 8 √ 2 = 2 √ 8, are capable of addition, substraction, multiplication or division; for they are being added 3 √ 32 that is √ 288; substracted, √ 32; multiplied √ 4096; divided √ 4: but such as are neither commensurable nor reciprocal cannot be amassed into one sum. And the sum of the former addition of √ 8 + √ 2 being already reduced to 3 √ 2 may be yet further reduced to √ 18 for 3 √ 2 is equal to the square root of three times three times two as hath been more than once shown. And generally when the Surds given are denominated by numbers in quadruple proportion, as √ 2 to √ 8, and √ 3 to √ 12, etc. the lesser and the greater twice being added together, as 2 to 16, or 3 to 24, the square Root of the sum is equal to the sum of the two square Roots given to be added, that is, √ 2 + √ 8 = √ 18, and √ 3 + √ 12 = √ 27. The reason is, √ 1 + √ 4 = √ 9, which 9 is composed of the lesser once and the greater twice, that is, as often as the √ 1 is contained in the √ 4. But if the numbers be prime one to another, they must be added or subtracted by the signs + and −, for these Rules reach not to primes. And having said this little to acquaint such as are wont to be afraid of operations where Surds are present, with this which will render some things easy which perhaps seemed hard, and others which were hard, less difficult. I will now leave this ragged Subject, and recreate a little with a few easy Propositions; the performing of which may serve to recall into use and practice that which hath been spoken of Solids in the former Chapter. CHAP. VII. Prob. 1. ANy right line being given, to divide it into two parts, so as the Rectangle of the whole and one of the parts; may be to the square of the other part, in such proportion as is betwixt any two right lines given. Let the right line given be b. The segment to be squared a. Then the other Segment is b − a. And let the two lines given be r and s. Then bb − ba′ aa″ r′ s″. And raa = sbb − sba. per 16.6. Euclid. That is, raa + sba = sbb. Make sb/ r = d, and divide all by r. Then it is, aa + da = db. Make db = ff. Then lastly it is aa + da = ff. And a is easily found by Rule 1. of Chaep. 2. And if it had been required to have had the Rectangle + or − some other plain to have had any limited proportion to the square aa, the work had been almost the same, with some small addition. Prob. 2. To make a Scalenon Triangle, of which the Base, Perpendicular, and proportion of the other Sides shall be given. (I account that the Base which subtends the divided angle.) Let the base given be b, the perpendicular c, and the proportion of the other sides, as r to s. Of which let r be the lesser. And for the lesser segment of the base put a: Therefore by supposition, So that the squares of them are also proportional, That is. rr′ ss″, cc + aa′ cc + bb − 2 ba + aa″. And by multiplying the means & extremes, It is, ssaa + sscc = rrcc + rrbb − 2 rrba + rraa. That is, ssaa − rraa + 2 rrba = rrcc + rrbb − sscc. Make ss/ r = x. And divide all the aequation by r, Than it is, xaa − raa + 2 rba = rcc + rbb − xcc Secondly, make x − r = f, & gg = bb + cc. Then it is, faa + 2 rba = rgg − xcc. Again make rg/ f = h, and xc/ f = k, and divide faa + 2 rba = rgg − xcc by f. Then it will be aa + 2 rb/ f a = hg − ck. Lastly, make 2 rb/ f = q, and hg − kc = mm The Aequation finally reduced will be then aa + qa = mm, and a may be found by the first rule for square Aequations. Chap, 2. Prob. 3. Any number being given, to find two other numbers, so as all the three may constitute a Rectangle Triangle. Unto the Square of the number given add unity, the half of the sum shall be the hypothenuse, or from the said square take unity, the half of the remain shall be the middle side. For let the number given be a, the square is aa, to which adding unity, the sum is aa + 1, the half whereof is ½ aa + ½, for the Hypotenuse. Secondly, from aa take unity, the rest is aa − 1, the half whereof is, ½ aa − ½ for the middle side. But the lesser side (by supposition) is a. The square of the lesser side is aa. The square of the middlemost is ¼ aaaa − ½ aa + ¼ Both those squares are ¼ aaaa + ½ aa + ¼: But the square of the hypotenuse, viz, of ½ aa + ½ is equal to these, that is, ¼ a4 + ½ aa + ¼. Therefore by the 48. of the first of Euclid the Proposition is proved. COROLLARY. Hence it is plain, that the two greater sides of any rectangle Triangle differ by unity, for if two Squares differ by 2, their halves differ by 1. NOTE. If it be required to have all the three sides in whole numbers, than the lesser side must be an odd number. Probl. 4. The difference of the sides of a Rectangle, with the Area and diagonal in one sum, being given in numbers, to find out the sides. Let the difference of the sides be 7 And the Area and diagonal together 73. And put the lesser side equal to a. Then the greater is a + 7. These two multiplied produce aa + 7 a, which is equal to the Area. And therefore 73 − aa − 7 a is the Diagonal The square of which is + 5329 − 146 aa + aaaa + 14 aaa + + 49 aa − 1022 a Which reduced and rightly ordered, Is + aaaa + 14 aaa − 97 aa − 1022 a + 5329 Which by the 47. of the first of Euclid, is equal to the two squares of the other sides a, and a + 7, whose squares are aa, and aa + 14 a + 49. That is, + aaaa + 14 aaa − 97 aa − 1022 a + 5329 = 2 aa + 14 a + 49. That is + aaaa + 14 aaa − 99 aa − 1036 a + 5280 = 0. That is, − aaaa = 14 aaa + 99 aa + 1036 a = 5280. In which aequation, because aaaa hath four dimensions, and the Homogeneal 5280, but four places, the root a cannot consist of more than one place, or figure, which must be found out by trying every one of the nine Digits, if need be, and will be found at last to be 5, therefore the other side is 5 + 7 = 12, the Area 60, and the diagonal 13. But if a had been more or less than 5 yet (except something else lead a readier way) it is good to try 5 at first, if it be too little then 7, if that also too little, than 9, so there will be no need to try the even numbers, 6, 8, etc. for if 5 be too little and 7 too great, it must be 6, the like reason will serve for 8, 4, 2, so that he which guesseth most unfortunately, needs not try above four or five digits, which is no great matter, the like happening sometimes in seeking the quotient in plain division, for no man is sure to guess right at first. But that we may exemplify this in bigger numbers, where a may consist of two or more places. Let the difference of sides be 71 The area and diagonal together 1177. Working as in the former example, there will arise an aequation, which being reduced and ordered as before, will be − aaaa − 142 aaa − 2685 aa + 167276 a = = 1380288. And putting b + c = a: Then the Canon of the resolution will be − bbbb − 4 bbbc − 6 bbcc − 4 bccc − cccc − 142 bbb − 426 bbc − 426 bcc − 142 ccc − 2685 bb − 5370 bc − 2685 cc + 167276 b + 167276 c. To be orderly subtracted from the homogeneal number given 1380288, as followeth. The number given The first single root b = 1. Subtract (the difference of + & −) +1252260 Remains of the number given The first Root decuplate b = 10. Then is all the − And is all the + Divisor is their difference. The second single Root c = 3 Remains of the number given All the − being 373800 The difference is +128028 Which being taken from the remains of the number given + 128028, there remains finally nothing, so that the given aequation is justly resolved by the Root b + c = 13. The lesser side a is therefore 13, to which if the difference given, namely, 71, be added, the middle side 84 is thereby composed. Again, if to that middle side 84 be added unity, the hypotenusa of a right angled triangle is composed, whose three sides are 13, 84, 85. The Superficies of this Triangle is half the parallelogram or rectangle required. For 84 multiplied by 13, giveth 1092 for the area of the rectangle, to which adding 85 the diagonal, composeth the number 1177, as was required in the Proposition. COMPENDIUM. Seeing the two greater sides of any rectangle triangle, exceed one another by unity (as by the former Corollary) the difference betwixt the two lesser sides being given, the difference betwixt every two sides is also given. So that putting a for the lesser side of the rectangle, the greater side is a + 71, and the diagonal a + 72, whose square is + aa + 144 a + 5184, to which the two squares of the sides, being aa + aa + 142 a + 5041, are equal: That is, 2 aa + 142 a + 5041 = aa + 144 a + 5184 And subtracting from each part aa + 144 a + 5041 There will remain + aa − 2 a = 143. And a will be found 13, by the second Rule of Chap. 2. RESUMPT. In the second Problem of this Chapter it hath been showed how upon a Base and Perpendicular and proportion of the remaining sides given, to describe a Triangle. It is there to be understood of an acute angled triangle, in which the perpendicular falls within the triangle. Now therefore let it be otherwise. mathematical diagram As in this figure, of the triangle adb, let there be given, the base ab = b, the perpendicular de = c, and the proportion of the other sides, namely bd = r, and da = s, whereas in the second Problem, the lesser segment of the base was called a, and the greater b − a. Now here the line be may be called a, and ae shall be b + a, and the rest of the work will be like that before in Probl. 2. NOTE. And it may be noted, that if from a and b (the ends of the line ab) be drawn other binary lines, how many soever, so as they hold the same proportion as s to r and concur in other points, as c, f, g, etc. Those points are all in the circumference of a circle whose centre is in the line ab, produced towards g. For upon ab describe the rectangle triangle abc whose two sides ac, cb, may be as s to r, and divide the angle acb into two equal parts by the right line cx, and draw cq perpendicular to ac in c, then the angle xcq = 90 − xca likewise the angle cxq = 90 − xcb, but xcb = xca therefore xcq = cxq; and cq = xq. And because of the similitude of the triangles acq and cbq it is aq′ cq″ bq′″ that is aq′ xq″ bq′″. Now by supposition ag′ bg″ ac′ bc″ And it hath been proved aq′ xq″ bq′″. Therefore by composition also, it will be aq + xq′ xq + bq″ xq′ bq″ But ac′ bc″ aq′ xq″. And xq′ bq″ aq′ xq″. Therefore ag′ bg″ aq + xq′ xq + bq″. And therefore ag = aq + xq. And xq = gq. But it hath been proved that xq = cq. Therefore xq = cq = gq. mathematical diagram That is ag′ bg″ xq′ bq″ therefore also By division ag − bg′ bg″ xq − bq′ bq″. That is ab′ bg″ xb′ bq″. Therefore the rectangles abq = xbg Eu:. 6.16. And because by supposition ac′ bc″ ad′ bd″. And ac′ bc″ qc′ qb″. Therefore ad′ bd″ qc′ qb″ but qc = qz, therefore ad′ bd″ qz′ qb″ and the angles abd, zbq being equal, the triangles abd, zbq are equiangular, Euclid. 6.6. And therefore ab′ db″ bz′ bq″ Eucl. 6.4. and the rectangles dbz = abq: Eucl. 6.16. But it was now proved that abq = xbg. therefore xbg = dbz. So that the points x, z, g, being in the circumference of the Circle zxg, the point d must be in the same circumference. Euclid 3.35. The like proof may serve to show that the point f is in the same circumference; which is all that was to be proved. This Circumference, however desired by the Ancients; and effected by modern Mathematicians, seems yet to have little use, more than to help the construction of the triangle, which (but now I shown) may be done without it. CHAP. VIII. Of Mixtion. DEFINITION. 1. STandard fineness, Is that fineness which the current Gold and Silver Moneys are made of. In England the Gold is 22 Carects of fine Gold, and two Carects of Alloy. The silver Monies are made of silver so as the pound weight contains eleven Ounces, two penny weight of fine silver, and 18 Penny weight of Alloy. DEFINITION 2. If any Ingot be finer than Standard, it is called better, if courser worse, and this betterness and worsenesse is reckoned by Carects and Grains in Gold, and by Peny weights in Silver, and is summed by multiplication of the betterness or worsenesse in the pound weight, or pound weights of the Ingot. DEFINITION 3. The Temper is that which when two or more quantities of Liquors, or Herbs, or Minerals, or any thing used in Medicine, of differing degrees of Heat, Cold, Drought, or Moisture, are mixed together, so as the whole Mass so made by mixing have none of these four Qualities. NOTE. The Standard and the Temper differ in this, the first respecteth but two qualities, to wit, better and worse: the lattet respects four qualities, namely, hot, cold, dry, and moist: yet the Mixtor dealing but with two of these at once, that is, such two as are opposite, as are the two first or the two last mentioned before, or any two which are alike, as both better, both worse, both hot, or both cold, may use the same way in both. Prop. 1. If there be two Ingots of equal weight, the one better than Standard by a certain fineness, the other as much worse, those two Ingots melted together shall produce a Fusion or Mass which shall be of Standard fineness Let the first Ingot b be better by c. The Second Ingot d worse by f. And let it be b = d and c = f. Multiply b by c it giveth bc equal to all the betterness of the Ingot b. By Def. 2. Likewise cf is equal to all the worstnesse, but bc = df. Therefore the whole Fusion b + c is as much better than standard as worse. Wherefore it is neither better nor worse, but just Standard fineness. Prop. 2 If two Ingots to be melted differ in weight, quality, and degree of quality reciprocally, that is, if as the weight of the first, to the weight of the second, so the degree of worstnesse of the second to the degree of betterness of the first, the whole fusion shall be Standard fineness. Let there be quantity b better by c. And quantity d worse by g. And let it be b′ d″ g′ c″. Therefore bc = dg Eucl. 6.16. Namely all the betterness equal to all the worstnesse, and therefore the mixture of the mass neither better nor worse. The same arguments will serve if the Propisition had been in Liquors, to prove the mixture to be temperate. Prop. 3. If there be two quantities of Silver or Liquor, of divers qualities, or divers degrees of the same quality, if all the betterness or all the worstnesse, all the heat, or all the cold be found out by multiplying each quantity by its quality, and taking the difference of them if they be opposite; or the sum of them if they be alike; that difference or sum divided by the sum of the quantity, gives (as some call it) the form resulting or the degree of betterness, worstnesse, heat or cold, of the whole fusion or mixture. Let there be quantity b hot in g. And quantity d cold in h. Then bg is equal to all the heat of b. And dh equal to all the cold of d. If bg > dh then bg − dh/ b+ d is the degree of the form resulting, Hot, Or if bg < dh, then dh − bg/ d+ b is the same in coldness. Now although this is plain from Def. 2. because all the heat bg − dh, or the coldness dh − bg of the whole mixture, ariseth by multiplication of the several qualities by their respective quantities, and therefore that whole heat, or whole cold divided by all the weight of the several quantities, gives the quotient equal to the degree of heat or cold of any part of the weights, which in respect of the whole weight may be called one, which degree of heat here bg − dh/ b+ d being multiplied by the whole weight namely by b + d gives bg − dh, that is all the heat of all the weight, and therefore bg − dh/ b+ d is that which we call the form resulting, and dh − bg/ b+ d if bg < dh. Yet this may be further confirmed by that Rule given by Mr. John Dee, in his Mathematical Preface before Euclid. The Rule which there he showeth is this. What proportion is of the lesser quantity to the greater, the same is of the difference between the degree of the form resulting and the degree of the greater quantity to the difference between the degree of the said form and the degree of the lesser quantity. Here therefore let be b < d for that is free. Also let it be bg > dh. It is to be proved by the said Rule, that b′ d″ h′ + Multiply the two later by the common denominator b + d, the first gives bg + bh, the second dg + dh. And therefore b′ d″ bg + bh′ dg + dh″ Multiply the means it gives dbg + dbh, likewise the extremes multiplied is dbg + dbh: And therefore the Analogisme which was to be proved is true, by the 16. of the 6. of Euclid. In like sort if it were bg < dh, and b < d, it might be proved, that Lastly, if g and h were like qualities, that is, both Hot, or both Cold, and b < d, It is then to be proved that And reduced b′ d′ bg − bh′ dg − dh″, Which is manifest. Example in Numbers. First, Let it be put b = 5, g = 4, d = 7, and h = 2. Then bg = 20, & dh = 14, & bg − dh/ b+ d = ½, Now because the heat bg is greater than the cold dh, the whole mixture shall be hot, and that Mr. shall be in the middle of the first degree, and according to Mr. Dees Rule it will be 5′ 7″ 2½′ 3 ½″, for the difference between the form resulting which is hot in ½ and the greater quantities degree, which is cold in 2, is 2½, likewise the difference between the lesser quantities degree, hot in 4, and the form hot in ½ is 3½: So that this is right. Secondly let be b = 5, g = 2, d = 7, h = 4, in opposite qualities h and g: dh − bg/ b+ d = 18/12, It will be 5′ 7″ 4 − 1 ½′ 2 + 1½ (for 18/12 = 1½. That is, 5′ 7″ 2½′ 3½″ as before, and the form resulting cold in 1½ degree. 3 Lastly, let be b = 5, g = 4, d = 7, h = 2, in like quality, for example both hot then bg+ dh/ b+ d = 2 10/12 for the form. And 5′ 7″ 10/12′ 14/12″ for the Anologisme, exactly agreeing in all cases with Mr. Dee. And this is so plain that it needs not be exemplified in metals, it not being my purpose to write much of them nor of the Standarding of Gold and Silver, because it is so neatly and fully done already in a little Treatise put forth in Anno. 1651. by Mr. Jobn Reynolds of the Mint. Yet the Reader may take notice that he which brings but common Arithmetic with him, may by some one or more of these three foregoing propositions, perform any plain Problem that can be required concerning mixtures in valuable metals or liquors. For first Rule. 1. If the weight of the mass be not limited, if any quantity with any quality (which exceeds not the degree of the greatest fineness) be given, a like quantity of the just opposite quality, will cause all to be Standard or Temper. Rule. 2. If the quantity of the mass be limited, and the two oppositive qualities given, then divide the quantity of the mass into two parts proportional with the qualities, and taking them reciprocally the mixture shall be Standard, or Temper, by the second Prop. Rule. 3. If there be two Ingots of Silver to be melted the first better by a certain difference, and the second also better not by the same difference, if each weight be multiplied by its betterness, the two products added together make the betterness of the whole mass; which being divided by the sum of the two weights, gives the form resulting of the mass by the third proposition, which mass may be made Standard by Alloy as followeth. As the fine silver in the pound Standard, Is to the form resulting: So is the weight of the mass, To the weight of the Alloy. But this Rule is not pertinent to the mixture of liquors, because in them there is nothing agreed on for Alloy. NOTE. If the two Ingots melted produce a mass worse than Standard, out of any Silver which is better, a quantity may be limited by the second proposition, to make it Standard But if there be given the weight of an Ingot worse by a certain difference; and the weight of the whole fusion be limited, and the fineness, wheher Standard, or better, or worse; This Rule doth it. Multiply the weight by the worstness, and divide the product by the betterness of the Silver to be added, the quotient shall be the weight of that to be added to make it Standard. And if it be required to have the fusion better, or worse than Standard (but not worse than the Ingot given) it is easily done by taking more or less weight of the fine silver to be added, or of more or less fineness as the case requires, and which needs no more than hath been showed. If the fusion consist of more than two quantities, all that hath been said of two things miscible, is appliable to other miscibles how many so ever, by repetition of the working with two at a time. Prop. 4 If there be three like solids equal in Magnitude, and differing in weight, the middlemost being composed of some of the matter of the first, and the rest of the matter of the third mixed, if the rectangle made of the weights of the first and third, Minus the rectangle made of the weights of the said first and second, be divided by the weight of the third want the weight of the first, the Quotient shall be equal to all the matter of the first (that is to the weight thereof) which is contained in the mixed solid. Let the first be b, the second c, the third d. And the weight of the first q, of the second r, and of the third s. And the common magnitude Unity, and make a equal to the weight of all the matter of the first contained in the second, as aforesaid. And make q′ 1″ a′ And s′ 1″ r − a′ Therefore a/ q is equal to all the said matter of the first, and r − a/ s to all of the third in the Mixture, I mean to the Magnitude of it. And Multiply all by qs (or first by q, and the product by s) it gives sa + qr − qa = qs, That is, sa − qa = qs − qr. But qs − qr is the Dividend required. And s − q the Divisor required. And qs − qr/ s − q = a by the last Aequation, And a equal to the weight of the matter in the first contained in the second, wherefore the Proposition is proved. Example in Numbers. Put q = 97 r = 73 and s = 63 Then qs − qr/ s − q = 28 9/17, which is all of the matter of b contained in c. And the residue 73 − 28 9/17 = 44 8/17 is all of d contained in c. Now 97′ 1″ 28 9/17 ′ 485/1649″ = a/ q And 63′ 1″ 44 8/17 ′ 756/1071″ = s − a/ 1 But . As it ought to be, the like proof serves for any Numbers. Prop. 5. If there be three like Solids of which the second is composed of divers matters, to wit of parts of the first, and parts of the third, and the three solids equal in weight, but not in magnitude, if the rectangle made of the Magnitudes of the first and third, less the Rectangle made of the Magnitudes of the first and second, be divided by the Magnitude of the third want the Magnitude of the first, the Quotient will be equal to all the matter composing of the first, I mean to the magnitude thereof, which is contained in the second Solid. The proof of this is the same with the former, Mutatis mutandis. Prop. 6 If there be three such Solids as before in the fifth, and the magnitudes of the parts composing found, if the magnitude of the parts of the first composing the second, be divided by the magnitude of the first, the quotient is the weight of those parts. For the common weight being Unity, As the first magnitude is to its weight, which is unity; So is the magnitude of the parts of the first Solid, to the weight of the said parts. (Not to retern the same form, but diffused in mixture, and compared in minute parts commensurable with the whole. Let the magnitudes of the entire Solids be f, g, h, and their common weight Unity, and let the magnitude of the parts of the first composing the second, be put equal to e. Then f′ 1″ e′ f″/ e which f/ e is the weight of the said parts composing. This is plain. Prop. 7. If there be three Solids, and the first and third composing the second as before, differing all in weight and magnitude: if the rectangle Parallelepipedon made of the weights of the first and third, and the magnitude of the second (all multiplied together) want the rectangle Parallelepipedon made of the first and second (I mean the weights of them) and the magnitude of the third (all multiplied together) be divided by the rectangle made of the magnitude of the first, and the weight of the third want the rectangle made of the weight of the first, and the magnitude of the third, the quotient shall be the weight of the parts of the first composing the second: which weight multiplied by the magnitude of the first and the product after divided by the weight of the first, this later quotient shall be equal to the magnitude of the said parts. Let the Solids be in Weight f, g, h, Magnitude b, c d, And f the weight of the first, and put the weight of the parts composing of the first equal to a. Therefore f′ b″ a′ that is, as the weight of the whole first to its magnitude, so the weight of part or parts of the said first, to their magnitude. Now because the weight of the parts of the first composing the second Magnitude g, are a, the weight of the parts composing of the third are therefore g − a. Therefore secondly That is, As the weight of the whole third, is to the weight of the parts thereof, so is the Magnitude of the said whole, to the magnitude of the parts thereof. So then the magnitnde of the parts of the first, more the magnitude of the parts of the third, are equal to the whole magnitude of the second. That is Multiply both parts by f h the rectangle of the denominators, it gives + hba + fdg − fad = fhc That is hba − fda = fhc − fdg But fhc − fdg is the dividend proposed. And hb − fd the divisor desired. And by the last equation. But by supposition a is equal in weight to the parts of the first composing the second, wherefore the proposition is proved, as to the first part. And the second part is manifest out of the first Analogisme to wit that ba/ f is equal to the magnitude of the parts of the first. Example in Numbers. Let the weight of the first solid be 85 Let the weight of the second 60 Let the weight of the third 54 The magnitude of the first 49 The magnitude of the second 50 The magnitude of the third 48 And the weight of the parts of the first a as before. Then And And therefore . That is the magnitude of the parts composing the second taken together, must be equal to the magnitude of the whole second. Multiply each part by 85 times 54, that is by 4590. It produceth 2646 a + 244880 − 4080 a = 229500 That is (rrduced) − 1434 a = − 15300. And a = 10 960/1434 Now the rectangle parallelepipedon of 85, 54, and 50, is 229500, from which taking 244800 (which is the rectangle parallelepipedon of 85 60 and 48) there remains − 15300 for the Dividend proposed. Secondly, If from the Rectangle of 49 and 54, which is 2646, be taken the Rectangle of 85 and 48, which is 4080, remains − 1434 for the divisor proposed. And (by the last Aequation) 19300/1434 = a, and therefore the Example in Numbers is cleared. In the Aequation before hba − fda = fhe − fdg, the quantity a is easily found by this analogism, namely, hb − fd′ hc − dg″ f′ a″, if one make hb − fd = mm and hc − dg = nn. And m′ n″ p′″, for then m′ p″ f′ a″. Upon the same way of reasoning which hath been used in this Chapter, is grounded the Rule of False Position, and also that of Alligation: For if the two degrees of the qualities of any two Miscibles, be called the two false positions, and the two respective quantities of the said Miscibles, be called the two Errors, than the degree of the form resulting is the true point sought. For if any one would work by the Rule of False, and go the nearest way, he must divide the distance betwixt the false positions into two parts proportional with the Errors, and the work is thereby done sooner than by the common way of Cross Multiplication. As if it were required to part 48 in two, and one of the parts again into 3, and the other into 4, so as the thirds of the one may be (in number) quadruple to the fourth's of the other. Suppose first 40 and 8, and dividing 40 by 3, quotient is 13⅓, and 8 by 4 is 2, whose quadruple should be 13⅓, but is but 8, so the first Error is − 5⅓. And putting the second time 30 and 18, the second Error will be found to be + 8. Make therefore 8 + 5⅓′ 40 − 30″ 8′ 6″ if this 6 be added to the second position 30 whose error we here worked with (namely with 8) the sum is 36 for the part required, and the other part is 12. And as for the Rule of Alligation, which is to add all the betterness and worstnesse of each particular component severally taken into one sum (which there is called the sum of the differences;) And then to work by this Anolagisme, viz, As sum of all the betterness and worstnes mixed, Is to the whole Mass, or mixture to be made; So is any particular betterness or worstness. To all that which is to be taken and mixed of that respective quality. All this being manifest, shall not need any proof. ☞ As before, Page 106, so here again, I let the Reader know that the word Magnitude in this Chapter, is to be taken for the number of small parts or atoms of a Body, and not for a line or Superficies. CHAP. IX. Of Mensuration. IN this Chapter I shall demonstrate little, as not intending to write much new, but (for the most part) such as hath been already exhibited by Archimede and others, yet put here because the Book should not want somethingfor the Reader which hath not read such Authors, and for such as stand in need of the thing rather than the Proof. mathematical diagram If there be a Cube whose side is bc, and a Sphere whose Axis we is equal to bc (which here we put 7,) and an upright Cone adf, whose base is df, equal also to bc, and its altitude ao = bc. 1 Then the superficies of the Cube (being equal to the Square bcfd multiplied by 6 is 294. 2 And the superficies of the Sphere (being quadruple to the Circle vaeo) is 154. 3 And the Superficies of the Cone (being made by multiplying the side ad = √ 61¼ by the semicircumference avo = 11) is 86, and not considerably more, to which adding the superficies of the Base 38½, the whole superficies of the Cone is 124½ 4 And if there be a Prism, whose Base and altitude are severally equal to the base of the Cube, or of any other rectangular Parallelepipedon, the Prism is the half of the Parallelelipipedon in solidity. Eucl. 12.7. 5 And if a Pyramid insist on the same Base with the Prism, having equal altitude, the Pyramid is two third parts of the Prism, or 2/4 of the Parallelepipedon. 6 The solidity of a Cone or Pyramid is found by multlplying its altitude by ⅓ of the area of the Base. The solidity of these other is found thus. For the Cube, Multiply the side [7] by the square of the side [49] it gives the solidity of the Cube, which is 343. For the Sphere, Multiply the Cube of the Diameter [343] by 11, and divide the product by 21, it makes the solidity of the Sphere, which is— 179⅔ The solidity of a Cylinder, whose Diameter and Altitude are the same with the Diameter of the Sphere, is made by multiplying the superficies of the base [38½] by the altitude [7] whereby the solidity is produced = = 269½ For the Cone, Euclid having proved it to be the third part of the Cylinder, Euc. 12.10. the solidity thereof is— 089⅚ A Fragment. The superficies of the Fragment of a Sphere is found by multiplying the superficies of the whole Sphere by the altitude of the fragment, and dividing the Product by the Diameter of the Sphere, and adding to the quotient the superficies of the base of the Fragment. The solidity of a Fragment (less than half a Sphere) is found thus. From the Semidiameter of the Sphere, subtract the altitude of the Fragment, and by the remain multiply the area of the Base, and subtract the product from that which is made by multiplying the semi-axis of the Sphere into the convex superficies of the fragment: Lastly, divide the residue by 3, the quotient shall be the solidity sought for. If the fragment be more than half a Sphere, subtracting this from the whole, the greater fragment is thereby had. This last Rule presupposeth the Axis of the Sphere to be known; but if it be not so, it may easily be found by the following analogy. Let the altitude of the fragment be b the semidiameter thereof c, Make and make f = cc/ b Than it is manifest by the 13 of the 6 of Euclid, that b + f is equal to the diameter of the Circle, or to the Axis of the Sphere. It is manifest by the former work, that the Solidities of the Cone, Sphere, and Cilinder, being 89⅚ 179 4/6 269 3/6 are in proportion one to another as 1, 2, and 3, for the Cone is ⅓ and the Sphere⅔ of the Cilinder, but the superficies of the Sphere and Cilinder are equal excepting both the bases of the Cilinder So by that which hath been said afore the Pyramid, Prism and Cube of equal base and altitude, are in solidity also as 1, 2, and 3. There may be other parts of a Sphere beside those which here are called Fragments, (not to speak of those which are irregular & Multiform) which are either Cones or Pyramids, whose bases lie in the superficies of the Sphere, and their Vertices at the centre, the solidity of one of these is found by multiplying the third part of the Base by the altitude, (which here is the semiaxis) the product is the solidity: These fragments are those which are usually called Solid angles. Example. Let there be a Pyramid of three sides, whose Base is 19¼, equal to ⅛ of the superficies of the Sphere, and the vertex thereof in the centre, it is plain enough that this Pyramid is the eight part of the Sphere in solidity. Multiply 6 5/12 (the third of the base) by the perpendicular 3 6/12, the product is 22 11/24, which is the solidity of the Pyramid, and multiplied by 8 gives 179⅔ equal to the whole Sphere. The like for Cones in this case. If the superficies of the Base be not wholly given, if any three things be given (if they be not the three angles) the three angles may yet thereby be found. And then, the Rule (which I had from my learned friend Mr. John Leake) is, If the excess of the three angles above 180 deg. be multiplied by half the Diameter of the Sphere, the superficies of any Spherical Triangle is thereby produced. mathematical diagram This may be thus demonstrated, let the superficies of the obliqne Triangle abc be required, from the greatest angle c, let fall to the base ab, a perpendicular cz, produced towards f, and continue the base ab till it cut zc, produced in n, and ac produced in o, than the triangle nfo is equal to acz, being equiangular, and having the sides cz and nf equal, for either of them is equal to the diameter of the Sphere, or rather a Semicircle want the line cn. Now if the angle at a be multiplied by the diameter of the sphere, it makes the superficies, acobz, likewise if the angle fco that is acz, be multiplied by the said diameter, they produce the superficies fnco. And both these superficies are equal to zbon + acz + fno, that is to a fourth part of the superficies of the sphere plus twice the triangle acz. But zbon is produced by the diameter multiplied by the right angle nzb, wherefore the diameter multipyed by the angles acz + caz is greater than the said diameter multiplied by a right angle by twice the triangle acz, and therefore the angles acz + caz − azc multipyed by the diameter, produce twice the superficies acz or by the semidiameter, they produce it justly once. And by the fame reason the angles bcz + cbz − bzc multiplied by the Radius produce the Superficies bcz, which added to acz make the whole abc. But the angles acz + caz − azc + bcz + cbz − bzc. Are the same as the angles abc + bca + bac − 180 d. which is the difference whereby the three angles given exceed two right angles. So that this excess multiplied by Radius as aforesaid, produceth the superficies of the whole Triangle at first given, namely abc, which was to be demonstrated. Many more such things might be taken out of Archimedes, as to measure the Superficies made by revolution of a spiral line, and others, which seldom occur to any vulgar use: And for that cause, and also because the recitation of them would not benefit the other sort of Readers which know them already, I meddle no further, but will leave this Subject after one short Rule for measuring Hogsheads or Barrels, which is this. From the area of the Circle of the greater diameter, multiplied by the length of the Vessel, subtract the area of the lesser, multiplied also by the length of the Vessel, and take the third part of the difference from the greater area multiplied by the length as before; the rest is the content sought for, in such measure as was the length, of the Vessel. That is Inches, if the Scale were so: of which 231 make a Wine Gallon, and 288¾ a Beer Gallon, or rather an Ale Gallon, according to some accounts, but not yet resolved fully. Otherwise thus. Let the square of the greater diameter in Inches be bb. The square of the lesser diameter in Inches cc. The length in Inches d. And let the content in Inches sought for be aaa. Then it will be Example. Let it be b = 6 c = 3 d = 10 Then 22 dbb = 7920 And 11 dcc = .990 In all 8910 Divide 8910 by 42 the quotient is 212 1/7, which is equal to aaa the content in inches, which was required. The very same number will come forth if one work by the former way, putting circumference to Diameter, as 22 to 7. But although this should be exactly true in one Vessel (which cannot be proved because of the irregularity of the Vessel) it would not be so in others, because of the irregularity (or diversity) of this irregularity. In Mr. Spidals Extractions there are many Propositions of worth, and all undemonstrated, I will therefore in this place bestow a Demonstration on one of the hardest of them, which is this. Let it be required to divide any Triangle, as cng from a point without the Triangle as q, into two parts, of which one part shall have any proportion to the whole, given between two right lines, as here the lines cd and cg. mathematical diagram By the point q draw qa parallel to the nearer side cn, cutting gc, produced, in a. Make aq′ cn″ ac′ cf″, And ac′ bc″ cf′″. And part cf into two equal parts in the point h, and draw the diagonal bh, And make he = bh. Lastly, draw the line qe. Then I say the Triangle cng is divided by the line qe into two parts cme and mnge so, that as cme is to the whole cng, so is cd to cg. DEMONSTRATION. Forasmuch as the right line cf is divided into two equal parts, in the point h, and to it is added another line fe, therefore the Rectangle cef + hfh is equal to the Square heh (by Euclid 2.6.) but he = hb, and therefore hbh = cef + hfh. But hc = hf, therefore hbh = cef + hch, and hch + bcb = hbh: Take away hch common to both Aequations, Than it is plain, that cef = bcb, because either of these is equal to hbh − hch: so than ce′ bc″ ef′″, Euc. 6.17. But ac′ bc″ cf′ ″ by construction, wherefore acf = cef, and seeing by construction it is aq′ cn″ cd′ cf″, therefore aqfc = den, & because ef′ cf″ ac′ ec″ therefore by composition ce′ cf″ ae′ ce″, but ae′ ce″ aq′ cm″, because the two last are parallels, Eucl. 6.2. And therefore aq′ cm″ ce′ cf″, and aqfc = mce, but also aqfc = ncd, as is proved before, and therefore the Rectangles ncd = mce, and also the halves of them are equal, namely the triangles ndc and mce, but the Triangles ndc and ngc have that proportion as have their Bases, cd and cg, wherefore mce′ ngc″ cd′ cg″, and the line me is drawn from the point q which was to be proved. I know this is demonstrated by others already, but I may aswell insert a Demonstration differing, as Mr. Spidall might write the same Proposition without proof. Some Corollaries might be deduced from this Proposition, by considering the various analogies therein, which, I leave to the invention of the Reader. Now if it were required to draw a line from a point given without a Circle given, through the Circle, so as to cut off an arch equal to an arch given, that may very easily be done in this manner. mathematical diagram From b draw the tangent br, and make br = c cd = b db = a Than it will be b + a′ c″ a′″, Euclid 3.36. Therefore aa + ba = cc, Euclid 6.17. Wherefore a may he found by the first Rule for plain Aequations, Chap. 2. CHAP. X. THe superficies of an Ellipsis may be easily found as near the truth as that of a Circle, because it hath been proved by divers to be a mean proportional between the two Circles described severally upon the diameters of the Ellipsis, and it is almost axiomatically evident by mere inspection of the figure following. And therefore it is as easy to give an Ellipsis in any proportion to another Ellipsis, as to describe any Ellipsis at all. mathematical diagram As for Example, Let the greatest Diameter of the Semiellipsis adc, be ac = 28. then the semicircle described thereon shall be abc = 88/2 and let the lesser diameter of the said Ellipsis be 2 do or fg = 14. Lastly, let it be required to describe an Ellipsis which should be to the Ellipsis adc, as 1 to 4. Upon the line ac from o both ways, set off foe and go, each of them equal to do, and divide do into two equal parts in h, then describe the Ellipsis which shall pass by the three points f, h, g, I say that the Ellipsis fhg is to the Ellipsis adc as 1 to 4. For seeing the Circle abc is to the Circle fdg in diameter double, therefore abc = 4 fdg, and of what parts soever abc is 16, of those fdg shall be 4. And seeing the Ellipsis adc is a mean betwixt them, the said Ellipsis is 8 of the same parts. Again, by the same reason the Circle fdg is quadruple to the Circle nhk. Therefore of what parts soever fdg is 4, of those nhk shall be 1. And seeing the Ellipsis fhg is a mean betwixt them, the said Ellipsis is 2 of the same parts. But the Ellipsis given adc is 8. And 2′ 8″ 1′ 4″, which was to be done. In like sort having duly proportioned the Diameters of Circles, may be made Ellipses, in any proportion one to another, or in any proportion to a Circle given. And the works may be proved by induction, as this also might have been, for seeing the circle abc = 616, the Circle fdg = 154, the Ellipsis adc, a mean betwixt them, must be = 308. Again because the Circle fdg = 154. And the Circle nhk = 038½ The Ellipsis fhg being a mean betwixt them must be = 77. But 77′ 308″ 1′ 4″, etc. NOTE. 1 Herein I make use of that proportion which is betwixt 22 and 7 for the Circle to the Diameter for easiness in account, small and whole numbers being also better attended and understood sooner by the Reader; and for no other cause: the more exact proportion being as 355 to 113, or (which is more used) as 360 to 114 5915492/10000000. NOTE. 2 Hence it is manifest that the Content of the Lunula adbc comprehended by the Circle abc and the Ellipsis adc, (being according to this account half the Circle abc, that is 308.) As also the mixed figures adf and cdg (being here the residue of the Semicircle fdg, to the Semiellipsis adc) may be found out as exactly as the Superficies of a Circle. with which, until a further discovery, we must be content. And I have here noted it, to show that investigation is not yet to be contemned, as if the thing sought were (not only impossible but) useless, when so many neat Propositions might thereby be started, as would (although not so absolutely necessary for present use, yet) delight the modest eye with the novelty. NOTE. 3 Moreover if the said Lunula adbc were composed of two Circles, there might be a rectiline figure given equal to the superficies thereof; That is, if the superficies of a Circle adc, were double to the superficies of the Circle abc, the lines ab, bc, being drawn, the triangle abc, would be equal to the Lunula adbc; as might be proved if it were not easy, and well enough known already. So that some figures of crooked lines, either differing in kind, or in quantity may be equalled with Rectiline figures, or numbers; And yet where Cirles of equal quantity include any Lunula or other figure, this cannot yet be done; So thin is that Curtain which is drawn between us and our desires. NOTE 4. Whereas in the former figure, the making of the Ellipses, adc, fhg; is not showed; this may be here useful to some: and it is as follows. The greater Axis of the Ellipsis being equal to the diameter of a Circle abc, namely, to the right line ac, the other Axis to be taken at pleasure according to the occasion; having here assigned the line do for the half of the lesser Axis, draw from the Circle to the diameter ac perpendiculars as many as you please. Then lastly, dividing each perpendicular into two parts proportional with bd and do, in certain points, if by those points (of which the more, the better) a line be drawn with an even hand, that line shall pass also by the point d and be the Ellipsis required. Otherwise, and more for Mcchanick use. Having chosen the two Axes ac, and 2 do and made them cut one another into two equal parts, and at right Angles in the point o; take the half of ac, and apply it both ways from the point d, to be Diameter ac in x and y, then in the points x and y which points x and y are called the burning points, fix two pinns, of Iron, or wood (as the greatness of the Plain shall require) And upon the Plane place a string that compassing both pinns shall reach just to the point d, or c, (for all is one) and there fasten the ends of the string together by a knot or otherwise, at which knot holding a Pencil, and carrying the Pencil round upon the Plane, about the pinns mathematical diagram with the string always strait, the Ellipsis (whose half is adc) shall be thereby described. Moreover (although I will not meddle much with this kind of Geometry) seeing these things are already richly treated in Greek and Latin, and not much more than named in any English Book that I have seen, I will write a little here of a Cone, and all the Sections thereof, comprehended in one figure, and after take some principal Definitions, and one or two ways of describing the Sections, and drawing tangents to them, and some few other Problems out of Claudius Midorgius, not word for word, but as it shall seem convenient here. CHAP. XI. Definition general of a Cone. mathematical diagram 1. Now let this Triangle abc represent half the Cone as aforesaid, and then if a plain, as eboaz touch the Cone all along from b to a, and make right angles with bc the diameter of the base, and again, another plain fd parallel to eboaz cut the semicone bac, the section it in the superficies of the Cone is half a Parabola, the other half underneath, if the Cone be supposed entire, and is not to be projected in plano. 2 Again, if the Semicone bac be cut by another plain gkz, parallel to the Axis axe, the section in the superficies of the semicone, to wit gk shall be half an Hyperbola, and the like for the other half underneath, if the Cone were supposed entire, and further, whatsoever plain cutting the Semicone as aforesaid being produced shall concur with the plain ba produced towards z. Thirdly, If the said Semicone be cut by a plain nph, neither of the former ways, nor parallel, nor subcontrary to the base, the line in the superficies, namely nh is a Semiellipsis. Subcontrary position is that where two like triangles are joined at an equal (and then vertical) angle, yet have not their bases parallel. Lastly, if it be cut by a plain lorq parallel to the plain of the base, the section or is a Semicircle. Definition 1. Opposite Sections are two Hyperbola's in opposite superficies cut by the same plain. Definition 2. The Vertex of a Section is a point in the greatest curvature thereof, but more generally the point where any diameter cuts the Section, and where the Axis cuts is called the highest Vertex. Definition 3. Any two lines applied within the Section, and equidistant, are called Ordinately applied, in respect of some diameter which divides them into two equal parts. Definition 4. Any line drawn so as it cuts the section, and divides the Ordinates' into two equal parts, is called the Diameter of the Section, and if it divide them as aforesaid, and at right angles, it is the Axis, and so much of the Axis or diameter as lies betwixt the Vertex and any ordinate is called (in respect of that ordinate) the intercepted Axis, or intercepted Diameter, and those two diametets which mutually divide lines applied in the Section and parallel to the Diameters, into two equal parts are called Conjugate diameters, of which, as likewise of the oppsite Sections, I intent to say no more in this Tract. Definition 5. The transverse Diameter of an Hyperbola, is a right line in the intercepted diameter continued without the Section, and is equal to the double of that line intercepted betwixt the Vertex and the centre, and connects the Vertices of opposite Sections: In an Ellipsis; or Circle, it is any whole Diameter: in the Hyperbola and Ellipsis, if it be the continuation of the Axis, or the Axis (in the later) it is called the transverse Axis. But the Parabola whose Diameters are all equidistant, hath no transverse Diameter, nor any centre. Definition 6. The Centre is a point where all the Diameters meet. Definition 7. The Figures of Hyperbola's, and Ellipses, and Circles are paralleligrams included between the transverse Diameter, and the contiguous Parameter, of whicb those are called transverse sides, and these Coefficients by some. Definition 8. The said Parameter is a right line drawn to touch the Section at the end of the intercepted Diameter, to which all the Ordinates' are parallel, and according to which they are compared, and valued, which is therefore called juxta quam possunt: and if it be contiguous to the Axis, it is called the right Parameter. Definition 9 The umbilicius, focus, or burning point in the Parabola, is a point in the Axis distant from the Vertex by a fourth part of the right Parameter. But in the other two Sections, the burning points are assigned in the Axis of either Section, distant from either end of the transverse axis by the space of a right line that is the square root of the fourth part of the figure produced by the said transverse axis, and the right Parameter, which applied to the transverse axis is in the Hyperbola excedent in the Ellipsis deficient. The same points in any Ellipsis whose diameters or diameter are given, may easily be found by the mechanic way of describing an Ellipsist a little before showed. Wherein also it is plain that these points are as it were Centre's proper to the generation of the Section. CHAP. XII. Of the description of the Sections. MAny are the methods General and Special, which Midorgius shows to describe these three Sections, I will only mention one or two. 1. To describe a Parabola about any Diameter given with one of the ordinate lines. Let the Diameter given be ab, and let bc be one of the ordinate lines applied unto it, by which the angle abc being given, join a and c by the right line ac, And let ab be divided into as many parts as you please, and through every such division draw right lines parallel to bc, and produce them, and make dk = √ bc in dg, likewise el = √ bc in eh, and fm = √ bc in fi, and so of all the rest, and the points c, k, l, m, a, etc. shall be all in the same Section, so that a line drawn with an even hand by all the said points, shall be by the first Prop. of the second of Midorgius, the Parabola required. mathematical diagram And bc on the one side, is equal to bc on the other side, because by supposition, that, and all the parallels to it kd, le, mf, etc. eaten those lines which are called Ordinates', or Ordinately applied, and so ab in respect of bc, also ae in respect of le, etc. are the intercepted Diameters, or if the angle abc were a right angle, the intercepted Axes. Def. 4. And if you make ad′ dk″ aq′″, and draw aq parallel to dk, than aq shall be the contiguous Parameter in respect of the intercepted diameter add, and so may the Parameter by ab, or any other diameter given, be found, and therefore the Parameter aq only being given, the Parabola by points may easily be described. 2. About any Diameter, and one Ordinate line, to describe an Hyperbola known in kind, in a plain by points. Let ab be a diameter of the Hyperbola, and bc an Ordinate to it, comprehending the angle given, abc, and let the Section be of such a kind, as that the transverse diameter to the contiguous parameter may be as r to s. mathematical diagram Make ab′ bc″ bd′″, And s′ r″ bd′ be″ and join the points d and e, and in the line ab take points how many soever, and by them points f, g, t, etc. draw lines parallel to bc, as fh, gn, ti, etc. the more the better, and making the triangle ded complete, produce these parallels both ways to the sides de, in the points h, n, i, etc. Lastly, making fk, gd, tl, etc. the square roots of the rectangles afh, agn, ati, etc. the points k, oh, and l, shall be in the Hyperbola required: per 5. of 2. Midorg. And therefore a line drawn with an even hand to pass by the said points, shall be the Hyperbola required. And the transverse Diameter thereof is the line ae. Wherefore, by the Proposition, if you make it r′ s″ ae′ ax″, than axe shall be the contiguous parameter to the intercepted diameter ba, supposing it drawn parallel to bc, fk, go; etc. Def. 8. And the lines fk, go, tl, are all of those which are called Ordinates' to ba, Def. 3. 3. About a diameter given, and an Ordinate, to describe by points in a plain; an Ellipsis known in kind. Let ab be any transverse diameter of the Ellipsis required, and cd one of the Ordinates' drawn at any angle given, as acd, to describe as aforesaid; etc. Make the rectangle be equal to the square of cd, and by a and e draw the line ae, and produce it to f, that is, so far as till it meets with bf, being made parallel to cd, and in ab take other points gh, through which draw lines parallel to cd. Lastly, to every Rectangle bgis, bhl, etc. mathematical diagram make squares equal, as the square of gk equal to the first, and of him to the latter Rectangle, and so as many as you please; the points m, k, d, shall be (by the 3. of the second of Midorgius) in the same Ellipsis of which ab is the transverse diameter, and bf the contiguous parameter, wherefore a line drawn with an even hand by those points m, k, e, shall be the Ellipsis required. COROLLARY. Hence it is evident, that having the transverse diameter of an Hyperbola, or an Ellipsis, the parameter contiguous is easily found by applying any Ordinate to the Diameter, as kg, and drawing a parallel to it from b, for making bg′ kg″ ig′″ a line drawn from a to i shall meet bf in the point f, so as it shall thereby determine the parameter bf in the Ellipsis, and the work (though not the letters) is the same in the other. RULES. In a Section given, the Diameter is found by applying two ordinate or equidistant lines divided both into halves, through which divisions the Diameter must pass. Def. 4. Secondly, drawing two other equidistants different in situation from the former, and dividing them as aforesaid, you have another diameter. Thirdly, produce both, and where they concur is the centre of the Section. Def. 6. Fourthly, Produce them still (in the Hyperbola) till the space betwixt the Vertex and the Centre be doubled, that doubled space is the transverse diameter: the Vertex is here meant at large, for that point of the Section through which the diameter passeth. Def. 5. Fifthly, Having the centre, an arch of a circle, any where within the Section, bisected, gives a point by which from the centre must pass the Axis. Midor. 1.54. I have showed already how the burning points may be found in an Ellipsis. Def. 9 Sixtly, In the Hyperbola let the transverse axis be b, the right parameter c produced till it make x equal to a mean betwixt them, bisect this mean in a, an arch drawn from a to the axis (the centre being the centre of the Section) shall there give the point desired. Midorgius 1.58. The burning point of the Parabola is obvious out of the 9 Def. To find the Axis of a Parabola. Seventhly, Because it hath no centre, but all the Diameters are parallels, find any one diameter by help of two ordinates', as aforesaid, and to it within the Section draw a perpendicular, which being produced both ways just to the Section, divide into two equal parts, and through the point of that division, draw a line parallel to the diameter found before, that parallel line is the axis required: The thing is so easy it needs no Example. CHAP. XIII. To draw a tangent to any point assigned in any Section, or from any point without the Section. NOt to trouble this little book with two many Figures, let the first Figure viz. of the Parabola be here resumed, which may serve by supposing the Diameter ab to be the Axis. mathematical diagram First let it be required to draw a line to touch the Parabola in the point m, and from m draw mf perpendicular to the Axis: produce the Axis ba, to z, making az = of and from z to m, draw the pricked line zm; the said line zm, is by 55 of the first of Midorgius the Tangent required. If the point m, had been coincident with the point a, a perpendicular to ba, in a had been the Tangent, per. 17 of the first Ejusdem. Now let there be a point given without the Section, (not in the Axis) at x, from which let it be required to draw a line to touch the Section. From x draw xp parallel to ab cutting the Section in some point, as here at m. And draw the Tangent zm, as aforesaid and make mp = mx And from p, draw a line parallel to zm, cutting the Section in r, and draw xr, than xr shall be the Tangent required by the said 55 of the first. Secondly, let it be required to draw a Tangent to any point in the Hyperbola dac, which shall be repeated here also, wherein let the Diameter of the Section ab be supposed to be drawn, and the line go any ordinate, and the point o to be touched by a right line to be drawn as follows. Having found the Centre y, as is showed in the former Rules, make yg′ ya″ yr′″, lastly from r to o draw the line roq for the Tangent required per ditto 53.1. And so by conversion of the work, if the point r were given without the Section in some Diameter or Axis, there might from thence be drawn a right line to touch the Section in some place, as here it doth at ●. NOTE. If the point to be touched were in the Section, and in the vertex a, then by finding another diameter, another Vertex comes in place, and a in respect of this other Diameter will be a point in the Section, and a tangent to it as easily drawn as to o: As may be seen in Fig. 2. Thirdly let the Ellipsis akb, be here represented, and let it be required from any point given in, or without the Section, to draw a Tangent. First in the Section at m. And from m to the other limb of the Section, draw here also any line as mm, and divide it into two equal parts in the point h, and finding the centre g, draw by g, and h, the Diameter ab and produce it, making gh′ ga″ gx′″: lastly, from x draw xm for the tangent required. Secondly, If the point x had been given without the Section, and required from thence to draw a Tangent to the Section in the point r, or where it falls, by conversion of the work. Make gx′ ga″ go′″ so have you the point o, from which a parallel to bf gives the point r, where a line drawn from x shall touch the Section. The working of these things in an Ellipsis is the same as in the Hyperbola, only seem unlike to them that consider not fully, because the centre and transverse diameter of the Ellipsis lies within, and of the Hyperbola without the Section. mathematical diagram And if h, or any point within a Section be given, and required through it to draw an Ordinate, that may be easily done, because it must be parallel to a tangent at the Vertex a. Any Section given, to find that diameter thereof, which shall make an angle with the Ordinate to it, equal to an angle given. If first the Section given be a Parabola, find any diameter, and from the end or vertex thereof, draw a right line to the Section, making an angle with the said diameter equal to the angle given, to which if a parallel through the middle of the other right line be drawn, that parallel is the diameter required. mathematical diagram Let there be given therefore the Hyperbola bac, and the angle z, to find the diameter egg, which with the Ordinate of shall make the angle ega = z. Find the transverse axis ad, and the centre e, and upon ad describe (by the 33. of the 3. of Euclid) a portion of a Circle dfa capable of an angle equal to z, then draw df and of, and through the middle of of draw egg the diameter required. The work is altogether the same in an Ellipsis, only the lesser axis is to be used. Midor. 3.67. Any Hyperbola being given, to find the Asymptoti. mathematical diagram Because ah, toucheth the Section, it is equidistant to the Ordinates', per Coral 2, ad 17 primi, But to the Rectangle or Parallelogram mag, that is to the figure comprehended of the two sides ma, and ag, is made equal the Square or Rhombus of ah, and an, is half of ah, therefore the square or Rhombus of an, is equal to a fourth part of the Square or Rhombus of ah, that is to the quadrant of the figure mag, and therefore by the 38. of the first and Coral. to it, by conversion it may be showed that the right line en, drawn from the Centre and produced how far soever shall never meet with the Section bac and by the same reason, and because an = ao, eo drawn from the Centre shall do the like, etc. From hence it appears, that the Asymptotes are lines drawn from the centre of the Section; and produced, so as that inclining toward the section still more, shall never be coincident therewith. More for the Parabola. Numerically. Let the base be given in Numbers 20, that is, of what length soever, let it be parted into 20 equal parts. And at any inclination to it, let there also be given a diameter; which divide into 100 parts. And through all the other 9 divisions of the Semi-base, draw lines equidistant to the Diameter shortening them in this proportion, viz. Of such parts as the Diameter is 100 let the next be 99, the next 96, the next 91, the fourth 84, the fift 75, the sixth 64, the seventh 51, the eighth 36, the ninth 19 A line drawn with an even hand by the ends of these lines shall be a Semiparabola. The said Numbers are made thus, 10 in 10. 11 in 9 12 in 8. 13 in 7. 14 in 6. 15 in 5. 16 in 4. 17 in 3. 18 in 2. and 19, in 1. Prop. 62. lib. 2. And they differ just as the square Numbers immediately succeeding to Unity, viz. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 &c, by the quantity of the odd numbers intercepted, as 1, 3, 5, 7, 9, 11, 13, 15, 17, etc. Which is the same proportion by which the degrees of Velocity of the falling of any thing toward the centre of the earth are increased, as Galileo hath sufficiently proved in his Dialogues. And therefore the course of every Projectile or thing shot from Gun or Bow may easily be proved to be a Parabolical line. And the making a Rectilone figure equal to a Parabola might be facilitated from hence, if it were not needless, the thing being already often done. Moreover it is to be noted, that the equidistant lines thus drawn, may represent squares, because they differ as the Square numbers do. For an Hyperbola, Numerically. The burning points and transverse axis being given, the Vertex is also given. Let the transverse axis be 80, the distance of each burning point 20 of the same parts, the said points a and b, the centre a space 23, and the other centre b, and space 103, describe arches, which shall cut where the Section is to pass, and so describing from the centre a other arches, 34, 57, 100, and from the centre b, with distances 114, 137, 180, other arches, so as the distances from b may exceed 100, as much as the distances from a exceed 20. Those arches of Circles shall intersect, and thereby give points by which the Hyperbola is to pass, by the 26. of the 2. of Midorgius. For an Ellipsis, Numerically. The burning points and Vertices being given (as they were before) the Ellipsis also may be described by numbers as followeth, let the one burning point be at a, the other at b, and let the diameter be z, the distance betwixt a and b let that be x equal to 100, and let it be x″ 16′ 100″, Therefore also z = 232, wherefore making the centre b at several spaces (more than 16, and less than 116, of such parts as z is 132) as 110, 97, 81, etc. describe arches. Again, making the centre a with distances 22, 35, 51, and others, still the correspondent compliments of the former distances to 132, draw other arches, which shall cut the former in points whereby the Ellipsis desired must pass, by the said 26 of the second. And it is plain from the generation of an Ellipsis, showed in the instrumental way before in this Book: for the string which describes it is always equal to z + x, that is, 232, and so is 100 + 110 + 22 and 100 + 97 + 35, etc. wherefore this is evident. And thus they that like this last way better, may accomplish the Section by number. Moreover, put the diameter of a Parabola of an inch ferè. And let the whole base (inclined to the Diameter at angle 84 ferè) be c = 150/64 Lastly, Let the perpendicular from the Vertex to the base be d = 64/64 Multiply 150/64 by 64/64 the Product is 9600/4096 Of which two thirds is equal to the supersicies of the Parabola, and is 6400/4099 Of these parts the middle Parallel which was before 75 (when the diameter was supposed 100) is 50/64 which doubled is 100/64 that is 6400/4096 as before. So that in this case the residue of the Rectangle or Parallelogram, when the superficial content of the Parabola is taken from it, and the length of the middle parallel, are both denominated by the same number, but this is left to the Reader to try by a Figure delineated by himself. But what use might be made of this (if it were further urged) either in natural or artificial numbers, I leave at this time also to the Readers inquiry and study. Here it may be noted, that a line being drawn to touch a Section in any point, if from that line in that point be raised a perpendicular, that perpendicular is said to cut the Section in the said point at right angles. For Example, let the right line ab touch the Ellipsis qpr in the point p, and let po be made perpendicular to ab in p, I say po is usually said to cut the Section qpr, at right angles. For, if any line drawn from the diameter qr to the Section, may divide it at right angles, let op be supposed to do so, and ab a tangent in p, as before. mathematical diagram Now if the angle apo be a right angle, all is proved, if not, draw the line qs by the point p, to cut po at right angles. It is manifest that qs shall cut the Section in p, because it cuts the tangent there. Wherefore the same right line po cuts two lines qpr, and qps at right angles in the point p, namely, where those two lines cut one the other, which is absurd. And into like absurdities will that opinion lead one which affirms that any crooked line can make any angle with any line whatsoever which toucheth that crooked line. For although Clavius against Peletarius and others may say that the angle of Contact (as they call it) made between a Circle and his Tangent is less than any acute angle made of right lines, yet seeing it is not divisible into parts aliquotas or aliquantas, which can have any other measure then the whole, that is, that each of them is less than any acute angle made of right lines, for this cause, and because it seems improper language to give the name of an angle to any space lying betwixt two lines, which although infinitely produced would never meet, I refuse to call it so. Which space being rejected as nothing, or at most non-angular, than the angle apo being a right angle, the angle qpo is equal to it, and is a right angle in any Circle, or Section of a Cone, or any other crooked line, how much soever composed. For herein it is the same with them, as with a Circle, namely, that in the point of any contaction, the angles on both sides immediately begun are equal. CHAP. XIV. I Shall here add a little to show the resolution of such Problems, which seeming to require two unknown points at once, are without help of a Conique Section (in lines) inexplicable. And other Problems may happen higher than these infinitely requiring four mean proportionals, or five or six. Or to divide an angle into five, seven or eleven parts, and appearing in Aequations of five, seven, or eleven dimensions, as need requires. I will begin with the most simple of these, Namely: Probl. 1. Betwixt any two right lines given, as r and s, to find two mean proportionals. Put a for the lesser mean required. And r < s Then rr′ aa″ a′ s″, Euc. 6.20. And aaa = rrs Eucl. 6.16. mathematical diagram Describe the Semi-parabola adg, so as r may be equal to the right parameter of it, which may be done by Chap. 12. hereof and in the axis ah make ac = ½ r, and from c raise the perpendicular ce = ½ s. And making the centre e, and Radius ea describe the circle fadx, cutting the section in d, and from d let fall a perpendicular to the axis ah in b, than the lines bd, ba are the two means required. For make r = 1, and bd = a, then because of the Parabola ba = aa, for the ordinate db is a mean betwixt the parameter which is unity, and the intercepted diameter ab, Chap. 12. And it rests only to be proved that s, or twice ce is equal to aaa, for 1′ a″ aa‴ aaa ' ' ' ' that is r′ a″ aa‴ sh' ' ' ' Draw dq, parallel to ba. Likewise draw de. mathematical diagram Then qe = ½ s − a, for qc = db = a. and ec = ½ s, likewise dq (that is bc) is aa − ½. the squares of which two, are, ¼ ss − sa + aa, and aaaa − aa + ¼. which together are aaaa − sa + ¼ ss + ¼. equal to the square of the: Eucl. 1.47. but de = ae, and the square of ae is ¼ ss + ¼. And therefore also it is aaaa − sa + ¼ ss + ¼ = ¼ ss + ¼. That is subtracting from each ¼ ss + ¼ the residue is aaaa − sa = 0. or aaaa = sa that is aaa = s, which was to be proved. This and all other solid aequations not transcending the biquadratique order, are explicable (as Des Cartes saith) by a small portion of any of the three Sections. Yet seeing he holds the Parabola the most convenient I make use of that also, and of his Examples in this former and the next succeeding Problem, aswell because a Parabola is much easier fitted to the work required, as also for that the demonstration thereby is not so anxious as by the other. Prob. 2 Now secondly let it be required, to divide any rectiline angle given into three equal parts, as the angle bag. Suppose it already done by the lines ae, of, and draw the chord bg, and also ec parallel to fa, lastly draw be. Now because of the similitude of the Triangles bae, bde, and ced, (for the angles are adg = aeb = bde = ecd) it it ab′ be″ de‴ dc' ' ' '. Put bg = b and bc = a. And let the Radius ba be Unity. Then de = aa and dc = aaa. And because bg + dc = 3 be. Therefore b + aaa = 3 a, or 3 a − aaa = b Which Aequation Pitiscus hath also in making of the Sins. Now suppose the Parabola akf drawn so as aq the right parameter may be equal to ba, that is Unity, and the part of the axis ac may be equal to ½, and ac = 2, then from e raise 'em perpendicular to the axis, and equal to ½ b, and upon the centre m, and the space ma, describe the Circle akftp, which shall cut the Parabola on that side remote from m, in two points kf, from which perpendiculars in g and d, shall be true roots of this Aequation, of which kg is the subtense of the third part of the arch required, and is equal to be, that is to a, and fd is the subtense of the third part of the compliment thereof to a circle, and if on the same side where m is, as from the intersection at p be let fall pl perpendicular also to the Axis, then pl, is a false root of this Aequation, and equal in Magnitude to both the true ones, that is pl = fd + kg. mathematical diagram But if in stead of − aaa + 3 a = + b, as in this Example it was, the aequation had been − aaa + 3 a = − b, than the true roots had fallen on the side where the centre m was, and the false root on that part remote from the said Centre. NOTE. This Aequation − aaa + 3 a − b = 0 is naturally without the second term aa, which is the cause that it hath the false Root not discerned by twice + or twice − succeeding, as hath been spoken of Chap. 4. If therefore one would have it so, he must fill up the second term, by augmenting the root never so little, putting e − x = a. The Demonstration of this Problem is as followeth. It is to be proved that kg, in the Section is equal to be the subtense of the third part of the angle given. Put kg = y. Then because of the Section. ag = yy. From the centre m, draw mk and ma, which are equal because of the circle. And draw kn, parallel to ae, and produce me to it in n. Then it is kn = ge = 2 − yy. The square therefore of kn, is 4 − 4 yy + yyyy. And mn, being equal to y plus half the subtense bg, call bg by the single letter, b, as before, then mn = y + ½ b, the square of which being yy + by + ¼ bb, add to it the former square of kn, that is 4 − 4 yy + yyyy, it makes + 4 − 4 yy + yyyy + yy, + by + ¼ bb, equal to the square of the Hypothenusal mk. Again the square of ae, is 4, and the square of me, is ¼ bb, which two squares are equal also to the square of mk, because mk = ma. Therefore 4 − 4 yy + yyyy + yy + by + ¼ bb = = 4 + ¼ bb. That is, − 3 yy + yyyy + by = 0. That is, (by adding on each part 3 yy, and subtracting yyyy) + 3 yy − yyyy = by. Or lastly (dividing all by y) + 3 y − yyy = b. But this aequation is alike graduated and like affected as the first aequation + 3 a − aaa = b. Wherefore y = a. But a = be and y = kg. And therefore kg = be. Which was to be proved. In like sort it might be proved that fd is a true root of the aequation 3 a − aaa = b (in the first figure) and the subtense of the third part of the compliment of the angle given bag to a Circle. And by such working one may find it evident that when a Circle cuts a Parabola in points, how many soever (the Vertex excepted) perpendiculars let fall from all those points to the axis, are all the several roots of one and the same aequation. Nor hath that aequation any more roots then those perpendiculars to the axis. NOTE. 1. In the aequation + aaa − bca = bbd, the construction differs somewhat from the former, for b being reputed unity, if c as here be signed with − the axis of the Parabola must be produced from the point c in the axis within the Section distant from a by ½ beyond the vertex, till the continuation be equal to ½ c, and at the end thereof raise a perpendicular equal to ½ d, at the end of that is the centre of the circle desired. And according to this method may any aequation, not above biquadratical be resolved, after by taking away the second term (if there be any) by the second Rule of Chapter the fourth it is reduced to such a form as this, aaa * bca = bbd, if the quantity unknown hath but three dimensions: or if it have four then thus aaaa * bcaa * bbda = = bbbf. Or else taking b, for Unity, than thus aaa * caa = d, and thus aaaa * caa, * da = f the signs + and − are hear omitted: for they must be supplied as the nature of the Aequation requireth. NOTE. 2. Note that in this breviate the line b, is that which was ba in the example of trisection, and that which was r or unity in the example of two Means: Also the line c is that which in the former example of trisection was 2 ce, or 3. And if this quantity be nothing, than the perpendicular mathematical diagram equal to half d, is to be erected at the end of half b, or ½ set off from the vertex upon the Axis within, but if c have any length, then at the distance of ½ c from that end, upon the axis. And this which hath been said is enough for all cubics. Prob. 3. But where the equation is aaaa − caa + da = f so placed as here, if there be + f and the Problem be to find the value of the root a, then producing ma towards a, Make as equal to the right parameter of the Section, and make axe = f, and upon the diameter xs describe the Circle xhs cut by a perpendicular to ma, namely ah in h, then making the centre m and the space mh, describe the Circle desired. But if it be − f, as in this Example I put it, then after ah is found as before: upon the diameter am describe a circle, and in it from a apply a line ai = ah, and making the centre m and the space mi describe the circle fik, which is the circle sought for. Now this Circle fik may cut or touch the Parabola in 1, 2, 3, or 4, points, from all which perpendiculars let fall to the axis give all the roots of the Aequation, as well the true as false ones. Namely, if the quantity d be marked − than those Perpendiculars which are on that side the Parabola where the centre m is, are the true Roots, but if it be + d, as here, the true roots are those of the other side, as gk and no, and those of the centre side as fz, pq, are the false. DEMONSTRATION. Put ce = c/ 2 and draw me perpendicular to ag, and gl equal and parallel to it: lastly, Put gk = a, then ag = aa, and taking from it ae, that is ½ c + ½, then goe = aa − − ½ c − ½, whose square is aaaa − caa − − aa + ¼ cc + ½ c + ¼. mathematical diagram And because by construction gl = ½ d, therefore kl = a + ½ d, and the square of it is aa + da + ¼ dd, which added a⁴ the former square of ge, it gives the square of km, that is a4 − caa + ¼ cc + da + ¼ dd + ½ c + ¼. Again, the square of ae is ¼ cc + ½ c + ¼. To which adding the square of me, that is ¼ dd the whole is the spuare of ma, to wit ¼ cc + ¼ dd + ½ c + ¼. But the square of ah, that is ai, is equal to f because sa = 1 and xa = f between which ah, or ai, is a mean. Therefore the square of mi is ¼ cc + ¼ dd + ½ c + ¼ − f But mi = mk. Therefore their squares are equal, that is. aaaa − caa + ¼ cc + ¼ dd + da + ½ c + ¼ = ¼ cc + ¼ dd + ½ c + ¼ − f. That is aaaa − caa + da = − f. or else aaaa = caa − da − f. which is the same aequation which was to be resolved, of which therefore gk, is a true root. In like sort might no, be proved a true root which was to be demonstrated. Des Cartes, demonstrates of all this no more but the case where the aequation is aaaa = caa − da + f, and leaves the Reader to please himself in finding proofs for the rest: I have chosen this case, to demonstrate, & have demonstrated the cases of the two Means, and trisection not only because some Readers may be as much pleased to have this done to their hand as left to do themselves: but also that all might see that the general way of demonstrating all sorts of cases, depends on these two things; first that the right parameter of the Parabola being always Unity, if any of the roots be put equal to a, the intercepted diameter will be always aa. Secondly, there may be ever found two squares equal to two other squares, and either the first two, or second two equal to the square of Radius. By help of these two things may any case hereof be proved. I will conclude with a Breviat of such equations as are not resoluble by Ruler and compass. 1 If there be as many vowels as consonants, and the vowels unequal. As, ae − damn = db, ae + da = db Or, − ae + da = db. 2 Though but of two dimensions and in fewest terms, as ae = bb, though such are solvable yet it may be by infinite ways, and therefore cannot be applied to any limited Proposition. 3 If there be but one Vowel, but cubically multiplied, or higher. As, aaa = bbc. Or, aaaa = bbbc. Where the Aequation being already in the least terms, and not to be brought down by any common divisor nor the homogeneal bbc, reducible to any solid more regular, as, to fff, it is irresoluble. CHAP. XV. HAving said (in the conclusion of the former Chapter) that the Aequation ae + da = db is (as by right lines and Circles only) irresoluble, I will here show a Problem, by resolving whereof the said aequation will be happened on, which is this following. Probl. 1. In any rectangle (bdca) given, from an angle in it [c] to draw a right line [cf] cutting one opposite side [bd in o] and concurring with the other [bq] produced in [f,] so as the intercepted line [foe] may be equal to [z] any other right line given. mathematical diagram Put bd = b ed = d do = a and bf = e And because the Triangles bof, do, are like, Therefore it is b − a′ a″ e′ d″, And db − da = ae, that is, ae + da = db. So we are quickly come to the Aequation required, which aequation having as many unknown quantities (as a, e) as known (to wit, b & d) is hitherto useless. That the Problem therefore may be solved, we must work another way, and bring it to a Solid Aequation, by making (for more convenience) cd = b. fo = c. and bd = d. and bo = a. Then d − a′ a″ b′ and , And the square thereof is equal to cc, by the 47. of the 1. of Euclid. That is, Multiply all by the denominator dd − 2 da + aa It makes ddaa − 2 daaa + aaaa + bbaa = = ddcc − 2 dcca + ccaa, That is, aaaa − 2 daaa + bbaa − ccaa + ddaa + 2 dcca = ddcc. Make bb + dd − cc = ff, than ff shall be signed + because hereby supposition it shall be bb + dd > cc. And the aequation will c, aaaa − 2 daaa + ffaa + 2 dcca = ddcc. Expunge the second term which is − 2 daaa, by the second Rule of the 4. Chap. And because the Rule is not fully exemplified there in the operosity thereof, I will here work it at large. Because aaaa hath four dimensions. Therefore make 4′ 1″ 2 d′ ½ d″ Again, because the first and second term have different signs, therefore put e + ½ d = a Chap. 4. Rule 2. The new Aequation arising thereof will be. The homogeneal − ddcc, is here put on the same side with the rest, because (for the present) it seems better to stand so, that it may be the last term, in relation to that which is gone before Chap. 4. Sect. 4. of the second Rule. In this last aequation, it is manifest that the second term 2 deee, is (through contradiction of + and −) abolished, as was required. And now because the quesititions root e must be found by help of a Parabola, as before in the like case was used, it is necessary to reduce the aequation to some such form as hath been showed before, in the Note of the former Chap. First therefore to reduce the third term, because d > f, and + 6/4 dd, taken from − 3 dd rests − 6/4 dd > ff, make 6/4 dd − ff = gg, so all of the third term shall be − ggee. Likewise for the fourth term, if + 4/8 ddd − 6/4 ddd + dff, be summed up together the aggregate will be − ddd + ffd, make dd − ff + 2 cc = hh, than all the fourth term will be + dhhe. Now for the last term − ¼ dddd − 1/16 dddd = 3/16 dddd and therefore making 3/16 dd − ¼ ff = ll, the aggregate of the last term is thereby − ddll, for ddcc is through contradiction of the signs annulled. And now the Aequation is eeee − ggee + dhhe − ddll = o, make gg/ d = m and hh/ d = n, and ll/ d = p, then the aequation will be eeee − dmee + ddne − dddp = o and making d = 1, than the aequation fully reduded and rightly prepared is + eeee − me + ne − p = o. (In reducing this or the like consider Chap. 5. Note. 2.) Or eeee = me − ne + p, which is altogether the same with that in the former Chapter, and the working of it is there showed. Except only because there the quantity f is signed −, and here the like quantity p is signed +, I shall (although this case only is demonstrated in Des Cartes) here demonstrate it thus. mathematical diagram Describe the Parabolaf ap, according to the parameter d, (that is as) & let ae be the axis, & make ac = ½ d, ce = ½ m, and at right angles at e make me = ½ x, and draw the line mas, making as = d, and axe = p, and upon xes as a diameter describe the semicircle xhs, and from a to h raise the perpendicular ah, cutting the Circle in h, and with radius mh describe the arch hky, cutting the Section in k, and from k let fall a perpendicular to me produced in q, and draw the lines mk and mh. DEMONSTRATION. To prove gk = e Suppose it done, and because kg = qe = e, and me = ½ n, therefore mq = ½ n + e, and the square of it is ¼ nn + ne + ee. And because ae = ½ d + ½ m, and gk = e, & because of the Parabola ga = ee, therefore also egg or qk = ½ m + ½ d − ee, & the square thereof is + ¼ mm + ½ md + ¼ dd − me − dye + eeee. That is (because d is equal to Unity) + ¼ mm + ½ m + ¼ − me − ee + e4 to which add the former square of ½ n + e, And then the whole is + ¼ mm + ½ m + ¼ − me − ee + eeee + ¼ nn's + ne + ee, That is, ¼ mm + ½ m + ¼ − me + e4 + ¼ nn's + ne equal to the square of Radius mk. Again, because me = ½ n, the square of it is ¼ nn, And because ae = ½ m + ½, the square of that is ¼ mm + ½ m + ¼, which added together make ¼ nn + ¼ mm + ½ m + ¼, for the square of ma, to which add the square of ah, that is p (for axe = p, and as = 1, and consequently by Eucl. 6.13. the square of ah is equal to p) the aggregate is ¼ nn + ¼ mm + ½ m + ¼ + p. equal to the square of Radius mh, but mh = mk. And therefore this aggregate is equal to the former aggregate & one may see that these quantities + ¼ mm + ¼ nn's + ¼ + ½ m. are common to both; And therefore the residuals are equal. Namely, + eeee − me + ne = + p, That is in Des Cartes his form. eeee = + me − ne + p, Which was the aequation to be resolved, and therefore gk = e, which was to be proved. And therefore it gk be added to ½ d, the sum is equal to a the quesititious root of the first aequation, and is equal to bo. So that the point o which was first sought, is hereby found; and the Problem satisfied; which was to be done. ADVERTISEMENT. Now that the different ways of writing equations may cause no confusion, let it be supposed to be written ever thus. eeee, me, ne, p = o. the signs + and − to be supplied as the occasion requires. Then, 1 If it be − m, the centre of the desired Circle is within the Section, or at least below the Vertex. 2 If it be + m, the said centre is above the vertex, and ½ m is applied upon the Axis produced, from the point c, which is always in the axis within the Section, distant from the vertex a by half the parameter; and therefore in this case the line m cannot be less than the parameter d, otherwise this point d, would still fall within the Section. 3 If it be + n, those perpendiculars (let fall from the several intersections of the Parabola and Circle to the Axis) which are on the same side with the centre are the false roots, and the other the true roots, but if it be − n, then just contrary. 4 Lastly, if it be − p, than one auxiliary circlewill serve, as here it doth, but if it be + p, then there must be another also, the describing of both which is showed in the former Chapter. Probl. 2. Upon a line given as a Base to describe an Isosceles triangle, so that an inward parallel Base may cut off two segments of the sides betwixt the bases, so that either segment may be equal to the inwad base, the perpendicular from the vertex to the said inward Base being also by supposition given. Let there be given the line cd, and the line bg. And let it be required upon cd, as a base, to describe the Isosceles bcd, so as the line bg, falling at right angles with fe, equidistant to cd, the lines df, of, and ec, may be equal each to other. Put gr = e, and fe = a, And cd = b and bg = d, Then d′, d + e″, a′ b″, mathematical diagram That is da + ea = db. Euclid. 6.16. which is the same aequation as was first in the former Problem; and therefore if there be in the room of da + ea = db substituted another equation like that in the former, such is the aequation. a4 − 2 daaa + ffaa + 2 dcca − ddcc = o. And that purged from the second term, as before, there ariseth a second equation. + eeee − me + ne − p = o And last, if d be a parameter, according to which a Parabola is described, the root e, and consequently the root a, may be found as in the former. And thus having showed the method general for all Aequations which attain but 3 or 4 dimensions, and exemplified it by Problems which lead to such aequations; I now say that was the end of my present business. And if any still desire a longer reach, I refer him to Des Cartes, who hath proceeded to aequations of 5. and 6. dimensions; by which four Mean proportionals, and quinquisection of Angles, and other sursolid Problems may be found and effected. Note 1. The first Problem of this Chapter, as it is more composed than trisection, so it comprehends it; as may be seen by (not only Pappus and others who applied it to that end but) the following Example. In which, let there be an arch of a circle, namely bc, and let it be required to divide the arch bc, into three equal parts, or (which is as good) to find the third part of the arch bc. mathematical diagram From o through e, draw oez. Now because eo is Radius, fp the diameter, and the angle fepa right angle, therefore the lines fo, po, eo, are severally equal. And the angles zeb = geo. and peo = epo. And also foe = 2 peo. Euclid. 1.32. Therefore also foe = 2 zeb. But foe, that is coz, being in the periphery, is measured by half the arch cz, Euclid. 3.20. wherefore bz which is the measure of half the angle coz, is a fourth part of the whole arch cz; and consequently a third part of bc, and therefore go which is equal to bz, is also equal to a third of bc, which was to be proved. NOTE 2. Mechanically. Seeing in this Scheme the line fo, is ever equal to the Semidiameter egg, if in common practice there is bc, or any other arch (not greater than a quadrant) to be trisected, laying a thin Ruler to touch the point c, and cut the diameter bg, produced in p, the point required, the compass being opened to the just length of Radius egg, setting one foot in ex, and shifting the Ruler till the other foot fall in the periphery at o, the point o, shall always be distant from g, by a third of bc, the doing of which (although it must not be called Mathematical, yet) is very near as easy, and as free from erring as from a point given to a point given to draw a straight line; or upon a centre given with a distance given to describe a circle; & from a given point in it to set off an arch equal to an arch given: And therefore I recommend this as the most simple and short and safe way for Mechanic use. NOTE. 3 If the arch to be trisected be greater than a quadrant, then trisect the compliment thereof to a semicircle; and the third of this compliment taken from 60 degrees (which is always a given arch) leaves the third of the arch required. Inscription of Chords in a Circle, and making aequicrurall triangles whose angle at the base shall be to the angle at the vertex in any given proportion, are the same thing: for to inscribe a figure of 3, 4, 5, 6, 7, 8, 9, 10, 11, sides &c, finds such triangles whose said angles shall be as 2/2, 3/2, 4/2, 5/2, 6/2, 7/2, 8/2, 9/2, 10/2, &c, as is easy to be seen by the operation. Quadrature of the Circle is that in which (as yet) only Archimedes hath laboured with any success; he having demonstrated that the Circumference is to the Diameter lesser than as 22, to 7, and greater than 21 70/71, to 7, within which strict limits, a French man many years since found it to be in whole Numbers. thus, As the whole circumference, is to the perimeter of the inscribed Square: so is 10, to 9, that is. Quadrant′. Subtense″. 10′. 9″. which is easy from practice; and may be proved. thus, Put the Diameter = 7, the half = 3½, of which the squar is = 49/4, and doubled is 49/2, whose square root is the side of the inscribed square. The whole perimeter therefore is 4 √ 49/2, and the whole circumference is found by this Analogisme. 9′. 10″.4 √ 49/2′. 40/9 √ 49/2″. It rests to be proved that 40/9 √ 49/2 is greater than 21 70/71, and lesser than 22. Now 4,949, is something less than √ 49/2 which multiplied by 40, and divided by 9, the quotient it 21, 995, which yet is greater than 21 70/71. for 995/1000 > 70/71. Again 4, 950. is something greater than √ 49/2, which multiplied by 40, and divided by 9, as before, the quotient is 22, so that 40/9 √ 49/2 < 22. and 40/9 √ 49/2 > 21 70/71. which was to be proved CHAP. XVI. Of Dialling. Probl. 1. Upon any declining Plane, to find the height of the Style, and place of the Substile, the Declination being first known. mathematical diagram Then if from the point y, to the stile, be let fall a perpendicular: and a line equal to it set off from y, towards a, it shall be the semidiameter of the aequator. Which aeqaator being described a line drawn from the centre thereof & produced to the point where the tangent gy would intersect ac, produced shall cut it into two equal parts, whereof that semicircle which is nearer to the line yg divided into 12 equal parts beginning at the said point of intersection of gy, & ac, produced & reckoning both ways as the Plane yq, shall admit, a Ruler laid from the centre of the aequator to every one of those divisions, shall intersect the tangent to the aequator, that is gy, continued through which intersections lines drawn from a, (the centre of the Dial) shall be the hour lines. For seeing it is known already in trigonometry that, as Radius to tangent compliment of the Elevation, so is the sine of the Declination to the tangent of the substiles distance from the Meridian, and if ac be radius, cd, is the co-tangent of Elevation, if that co-tangent be made Radius, it is manifest that the sine of the declination sir, shall be made thereby the tangent of substilar distance, yc. In like sort it might be proved that, as Radius cd, to cd the co-sine of the Elevation adc, so rc the co-sine of declination, to rc the sine of yag. Probl. 2. Of Declining Recliners. To find the Meridian of the Place, the Meridian of the Plane, and height of the Style. Let the parallel lines qs, lg, represent the base of the declining reclining plane; and make the angle gxn equal to the reclination, And let qs, and xw, cut at right angle in a. Make the angle xab, equal to the declination. And draw the lines xn, and ab. And making xc = xn, draw cb, for the Meridian of the place; or hour of 12. Then make am = ab, and the angle amz, equal to the latitude, and draw the line mz. Make ak = axe, and ad = an. Draw the line kd, and (where is need) produce it. mathematical diagram From z, let fall a perpendicular to kd, in h, and from b, (the centre of the Dial) to the vertical Meridian xw, apply br = kh, for the Substile. Lastly unto the line br, in the point r, raise a perpendicular re, = zh, and draw a line from, b to e, for the Style. So the line re, produced both ways, shall be a tangent to the Aequator, whose semidiameter shall be a line let fall from the point r, perpendicular to the line be. And the rest of the Dial may be finished like a Vertical decliner, as in Prob. 1 DEMONSTRATION. First the Centre b, is chosen at pleasure, in any place of the Horizontal Meridian ab, for the Parallels qs, lg, might be put nearer or further off and not alter the work at all, only the Diagran would be lesser or greater accordingly. Now because xc = xn, the face of the Reclining plane xn, shall cut the Vertical Meridian xw, and let it cut it in the point c. And because the line bc, subtends both the Meridian's ba, and ac, and is drawn upon the Plane from the Centre, it shall represent the Meridian of the Place: for the Sun enlightening the point c, at 12 of clock in the vertical xw, and the point b, at the same time in the horizontal meridian ab, it shall at the same time illuminate the whole line bc, so as the Style of the Dial shall shadow the same justly at the same time. Again, because the triangles anx, akd, are aequiangular and equal, if the point w, be the Zenith, the point x supposed to be laid upon the point k, the lines kd = xn, and the angle akd equal to the reclination, kd shall then truly represent the reclining Plane. Moreover am being equal to ab by construction, and the angle amz equal to the latitude or elevation, the point z represents a Pole of the Aequator, because the Axis mz, and the Meridian az meet there. mathematical diagram If therefore from the Pole z, a meridian zh fall upon the Plane kd, (or the back side thereof) at right angles, it shall fall upon a point of the substile in h, which point h therefore doth limit the substile on that part. But the horizontal line in which b, the centre of the dial is taken must limit it on the other part, to wit in the point x, or k, so that hk is the just length of the substile upon the Plane. And because the substile must both pass through the Centre b, and incline to the vertical xw (to which the plane itself inclines) the line br = kh being so placed is the substile. Lastly, because zh is the nearest distance betwixt the Pole and the Plane, ere being equal thereunto, and perpendicular to the Substile, it shall be the length of the Style, that is (where the Substile br, is Radius) the tangent of the height of the Pole above the Plane, which is all which was to be proved. This Plane declining 30 degrees and reclining 20 degrees, may serve for an example of planes reclining and declining which lie between the Zenith and the Pole. The distance of the Meridian from the Horizon is the angle gbc = 78. 50′. NOTE. Unto any Declination given may be found a Reclination, so as that the plane composed shall have the Aequator in the Zenith: that is the Poles shall have no height above the Plane, only by making an = az in the former figure, and drawing xn; the angle gxn is the thing sought. Or if the Reclination gxn, were given, a Declination might be found to do the same, only by making still az = an; and the angle azm equal to the height of the Aequator above the Horizon; and ab = am; and drawing the line ab, the angle xab, shall be the declination sought for. And those Planes so made are called meridional Planes, because, they are parallel to the Axis of the Aequator. The Dial upon any of them is like the Polar Plane, having the same reason for the parallelisme of the hours; but because the Substile may chance to fall neither upon our 12, nor 6, but any where else according to several declinations, the place of that must be found. Which is done by finding a certain angle comprehended betwixt the Meridian of the place and the Meridian of the Plane, which angle is commonly called the angle of inclination of Meridian's, and may be found as follows. mathematical diagram s. c.pz + Radius = tc. gzp + tc. gpz. And therefore adding the sine of the elevation to Radius, and from the aggregate subtracting the tangent compliment of the declination, the remain will be the tangent compliment of the angle of inclination of Meridian's, which angle sought is zpr for gpr is 90 d. Example. Let the Declination wzq be 60 degrees. The Reclination zg, so much as may cause the Plane qgx to pass through the Pole at p, and let pr be supposed to pass by the pole of the circle qgx, that so the angle rpg, may be a right angle: and choose pz for the middle part in the triangle bgz, right angled at g, because in the triangle gxz, the sides xz and xg are quadrants, then in latitude 51. 32′. The sine compl. pz, is the sine of the elvation. That is sine 51 32′ log. 9893745 To which add Radius, sum is 19893745 From which take tc. gzp that is, t. 30d.9761439 Remains tc. gpz or tang. 53d. 36′. = 10132306 The angle fpz, therefore is 53. 36′. Which is the angle of inclination of meridians sought for, which being divided by 15 (the number of deg. of the Aequator accounted for every hour) the quotient is 3 57/100, or rather 3 172/300. which shows the hour from Noon over which the stile must hang that is in the after noon 3 172/300 of clock if the declination be westerly, or 8 128/300. if easterly, this last the morning hour, the other afternoon. Now this fraction being rectified (which every man that hath any skill in Arithmetic knows how to do) and the entire hours of 8 and 9, or 4 and 3 being assigned, the rest may be found by tangents of arches increasing still from the substile by the quantity of 15 d. for every hour. And so the Dial may be made on paper. But to place it right after it is made, the angle comprehended betwixt the substile and the horizontal line which is here the line lg upon the Reclining Plane, must also be found out: And may thus (working still by Logarithmes) from the sine of the latitude plus Radius, take the sine compliment of the Reclination, there shall remain the sine of an angle, which angle is the true distance of the substile from the Horizon. And must be set off from the horizontal line at the centre of the dial westward if the declination (as here it is) be eastward: Or else eastward if the declination be westward. And so the dial shall be rightly made and situate. Now though all this be most easy to all that know how to use the Logarithmes; yet that this may not depend thereon, the same things may be found out Geometrically by describing any Circle at pleasure. For first, Cotangent Declination′. Radius″. sine latitude′. tangent Inclination of Meridians″. Secondly, Cousin Reclination′, sine Latitude″; Radius′. sine distance of substile from Horizon″. Which is enough, any circle will afford these naturally; for such as affect not the artificial, and the former Scheme will demonstrate this easily. Prob. 3. To find the same in those called South Declining Inclining Planes. Put the parallel lines p, q, for the horizontal Base of the inclining Plane. The Declination, angle abc = 60 d. The Inclination pcd = 58 d. Make yz perpendicular to the line q in b. Make also ce = cd. And bdz equal to the Compliment of the elevation. Join b and z. And make bk = bz. Through k (the line ab being first produced so far) draw ku parallel to zy, cutting the line qd in s. And make bl = sk. mathematical diagram From l let fall a perpendicular to cd to wit lo. Make hg = co. Then draw egg for the substile. And, Make gn = lo, and at right angles with egg. And draw en for the Style. And ea for the hour of 12, So the Dial may be finished like a Declining Vertical. DEMONSTRATION. Because abc = kbz, is equal to the declination, therefore the line kba, is the horizontal Meridian upon the base pq. And bk = bz by construction, and the triangles dbk and dbz are equiangle and equal, having one common side db; and a common angle bdz, and bk = bz, therefore a right line passing from k, in the horizontal Meridian to d, or e in the vertical Meridian zy, shall represent the Axis of the Aequator; for the angle bdk equal to bdz, is the compliment of the elevation by Construction; and dbk a right angle. And therefore whensoever the poin k, is either shadowed, or enlightened, the point a is the same; and the point e also, because it is in the same Axis with k, is at the same time so affected; wherefore the centre of the dial being at e, a line drawn from thence upon the Plane to a, shall be the hour of 12. And because the Hypothenusa dk, or eke, is the Axis passing from e, in the inclining Plane by k, in the horizontal Meridian; and the point k being in the line kshg, a perpendicular let fall from thence to the Plane shall fall in the same line kshg. Make ex = hg = co. Then a perpendicular from l to x, is the same with that from l to o, namely the line lo. mathematical diagram And because bl = sk. and cx = hg, therefore a perpendicular let fall from k, to g, shall be equal to lx, or lo. And because k (as hath been showed) is a point in the Axis, and g, a point in the Dial Plane, the perpendicular kg, or lo, shall be the height, or (as some call it) the length of the Style; or more properly (making egg, radius) the tangent of that height whereby the Pole is elevated above the Plane. And the point g, where it cuts the Dial plane at right angles, shall be a point in the substile. But e, or d, is the Centre of the Dial, as hath been showed already. Therefore a line drawn from e, to g, that is the line egg, is the Substilar. And because gn = lo, and the angle egn, is a right angle, therefore the line en, drawn by the points e, and n, is the Style. Enough is already written to show how to find the Meridian, Substile, and Style, in all declining verticals, and declining Horizontals, I mean declining and reclining Planes; In which there is this of brevity, that not only the things, that is their Magnitude, but their places and Situations upon the Planes, are all obvious together in the very working; or with a little transposition made so. I meant not to be general in this subject which is the reason why I have forborn to say any thing of Horizontals, Prime Verticals, Equinoctials, and Polar Dial's. Yet because the book shall not be rendered (to some persons) useless for want of these; at the end hereof I purpose to append a Table (out of Kercherus) by which the Horizontal and Prime Verticals all over Europe especially, may be made by the quantities of their Arches set down in the table and to be set off from the Meridian of the plane upon the Dial Plane, and may be measured upon any Circle there described at pleasure. Aequinoctial Planes (I mean such as are so denominated from the planes not the Poles) are such as have one of the Poles for Zenith, upon these, a circle divided into 24 equal parts gives the hours; and a pin perpendicular to the centre, (of any length) is the Style. Polar Planes have the Aequator in the Zenith; where these are proper Orisons, the Substile and Meridian of the Place are all one, the hours parallel to it; their distances from it measured by tangents of 15, 30, 45, 60, 75, 90, &c, degrees according to any Radius; provided that Radius be the same with the height of the stile; which is a pin set upright in the centre of the Circle to which the tangents belong. East and West Planes in any latitude are of this kind, differing only in longitude, 90. degrees; by which it comes to pass that in these the hour of 6. is the substile, and the rest of the work (leaving out unnecessary hours) the same as in the former. To place these right upon the plane draw a line parallel to the horizon; and in any convenient place from the horizontal line in an arch of some circle set off the latitude; included between the horizontal line & another, that other line, cut by a third line at right angles, the third line shall be the Aequator, the second the substile, the rest like that before. Prob. 1. Probl. 4. To draw any Vertical Dial by help of an Hozontal dial, without any Aequator. Making the Centre x, and at any distance describe a circle, on which (having made xa the Meridian) set off the horizontal Arches proper to the latitude (taken out of the table hereafter following, or any other way;) from the Meridian making thereby marks in the circle, by which, and the centre x draw the horizontal lines x 1. x 2. x 3. etc. on both sides the Meridian to cut the declining plane (which in this example is the line rt, declining from the Prime Vertical os, as much as is the angle oar that is almost seven degrees) in the points 1, 2, 3, &c, on the one side; and 8, 9, 10, and 11, or as many of them as the plane will receive on the other side of the Meridian: to which points lines drawn from the centre of the dial, that is z shall be the hour lines, the angle axp, is the elevation. The said centre z, is found out by making, az = ap, the said ap being always the sine of the latitude, where the line axe is the sine of the compliment thereof, that is, having made xa the meridian, and os, the prime vertical, and the angle axp equal to the latitude, the point p, and the line ap, are thereby given, then having made za, perpendicular to rt, make za equal to ap. Now to find the stile and substile, it is already showed at first in the vertical decliner. mathematical diagram Before I meddle at all with these, it will be necessary to proportion the Perpendicular Style to the Plane. CHAP. XVII. Of the Perpendicular Style. THis must be perpendicular to the substile, and the top thereof determined in the Style, or axis. If the Plane be small, consider whether it be direct, or declining, and much declining. If direct, the substile may be placed in the midst, if declining, then on the part opposite to the declination. The substile well placed, (and room left for the figures) divide it into two parts, so as that part next the centre of the Dial may be the tangent compliment of the height of the Pole above the plane; and the other part the tangent compliment of the Sun's meridional height in the beginning of that Tropic which is to be more remote from the centre of the Dial. And the Radius proper to these tangents shall be the perpendicular stile, to be placed in the point of Division in the substile, perpendicular thereunto. Of the Signs, or Parallels. A Sign is a twelfth part of the Ecliptic, and contains therefore 30 degrees. A Parallel, according to the vulgar sense, is the Sun's diurnal Motion day by day: And because there are 47 degrees from Tropic to Tropic, there may be so many Parallels, that is, circles which the Sun describes every 24 hours supposed Parallel to the Aequator though not exactly so; and although there are 47 of these yet in our latitude of 51. 32′. we account but 9 viz. those which are the day from Sun to Sun when it is 8, 9, 10, 11, 12, 13, 14, 15, or 16. hours long. The Description of these parallels and of the signs is made the same way: only due respect must be had to the quantity of the Sun's declination, for in all direct horizontals, the perpendicular stile being Radius, the tangent compliment of the Sun's height, in any sign or parallel, at any hour of of the day, set off from the foot of the said stile, and extended to the hour line, gives a mark, by which the parallel of that day shall pass. So that this Work repeated so often as the number of parallels to be inscribed, and the hour lines require, shall give respective points enough in each hour to draw each parallel by. Example. In the latitude 51. 32′. the Sun being in Pisces (the beginning thereof) the degrees of the Sun's height above the horizon at every hour being as followeth, that is, 25. 37′. at one of clock, 21. 49′. at two; 15. 57′. at three; 8. 32′. at four, and the same for eight, nine, ten, and eleven respectively, if the perpendicular stile being Radius, the tangents of the compliments of 25. 37′. 21. 49′. 15. 57′. 8. 32′. be applied from the foot of the stile to the respective hour, that is, the co-tangent of 25. 37′. from the foot of the stile to the hours of 1. and 11. and so the others, they shall give points in every hour-line one, by which a line being drawn with an even hand shall be the Parallel at the beginning of Pisces. And the like of all the rest. And therefore generally in verticals, as also in all recliners that is to say upon all planes whatsoever, draw a horizontal dial proper to the plane, and inscribe the signs or parallels upon it, by setting off from the foot of the perpendicular stile, the tangents compliments of the Sun's height at every hour in the beginning of every such sign, above that plane taken as an horizon, the perpendicular stile being ever Radius; and at the end of these tangents so set off, upon every respective hour-line, will be a point, by which points, lines drawn with an even hand, shall give the parallels desired. This horizontal Dial being drawn in obscure lines, the Dial for the plane may be drawn afterwards. The Parallels serving which were drawn before. Example. Suppose (as M. Wells doth pag. 185) a plane declining 30 degrees, and reclining 55 degrees; the height of the Pole above the plane 19 degrees 25 minutes; the Sun's height at the beginning Of Taurus to be at the hours of 12h. 82d. 5′. Of Taurus to be at the hours of 1 73 30 Of Taurus to be at the hours of 2 60 3 Of Taurus to be at the hours of 3 46 1 Of Taurus to be at the hours of 4 31 53 Of Taurus to be at the hours of 5 17 47 The tangents of the compliments of 82, 5′. and 73, 30′. and 60, 3′. etc. set off from the foot of the perpendicular stile (the said stile being the Radius to those tangents) to the obscure horizontal hours of 12, 1, 2, etc. give the true distances between the foot of the stile and those auxiliary hours for the parallel of Taurus. And so the other Parallels may be found. It is true, the height of the Sun at every hour of the day, at the beginning of every sign in any latitude is not easily found out without Trigonometrical Calculation by Logarithms of the sins & Tangents, or by trusting to Tables already Calculated, if any happen to be done for that latitude already, the way of making a table shall be showed towards the end. Of the Vertical Circles. These are vulgarly called Azimuths; and are great Circles whose Poles lie in the horizon, and intersecting one another in the Zenith and Nadir of the Place. The whole Horizon being divided into 32 parts equal, these circles showing those divisions are called points of the Compass, and marked S. SbE.SSE. etc. Every one distant from other by, 11¼ degrees. But the better way of accounting them is 10, 20, 30, 40, 50, 60, 70, etc. degrees from the Meridian. 1 In all horizontal dials, the Perpendicular stile being chosen, making the foot thereof the centre, at any convenient distance describe a circle, and account from the meridian both ways arches equal to 10, 20, 30, etc. degrees, from which divisions right lines drawn to the-foot of the stile aforesaid, shall represent those Azimuths upon that dial. 2 Upon a Prime Vertical (or South) Dial, through the foot of the perpendicular stile draw a right line parallel to the horizon, and making the said stile radius, upon the parallel line set off both ways from the Meridian tangents of 10, 20, 30, 40, etc. degrees, through which divisions right lines drawn all at right angles with the parallel line shall be the Azimuths. 3 Upon any declining Vertical the same being done shall give the Azimuths of 10, 20, 30, etc. from the meridian of the plane, or from the Meridian of the place, just allowance being made for the distance of Meridian's. 4 In South Declining Reclining Planes, the perpendicular stile being chosen, and made the Radius, the tangent compliment of the Reclination applied from the foot of the said stile to the meridian of the place, shall determine the Zenith of the place, through which, and the foot of the stile, that is the Zenith of the plane, a right line drawn shall be a perpendicular to the horizontal line, which shall concur with the aequator in the hour of 6, and the therefore if from the foot of the stile upon the said perpendicular towards the North (for the former application is made towards the South) be set off the tangent of the reclination, a line drawn from the end thereof at right angles with it, shall be the horizontal line: upon which the tangents of 10, 20, 30, &c, (the secant of the reclination being now made Radius) set from the said right angle, lines drawn from them to the Zenith of the place shall be the Azimuths. 5 The distance betwixt the Meridian's being known, upon the horizontal line, the Azimuths which were accounted from the meridian of the plane may be fitted for account from the meridian of the place with easy. For example, let that distance be the tangent of 20 deg. than that Azimuth which is 10 from the one, is 10 from the other also, and that which is 30 on the same side of the substile, is 10 on the other side of the Meridian of the place, the like Method serves for any distance. Note 1. It may be noted, that although I have showed the construction of a South reclining plane at the beginning hereof in a figure proper only to those planes which recline not further than the Pole, whereas in those that do, and although there be some variation of the Scheme as you may see by comparing this with the former (at the first beginning of this subject) for the point h, which mathematical diagram there fell on that side of the vertical meridian zx towards q, here falls on the other side towards s, likewise the hour of 12, that is bc, did there fall betwixt the axis and the substile, but falls here betwixt the substile and the horizontal meridian ba: yet this notwithstanding the construction is the very same in both. NOTE. 2. It may be further Noted, that as the Reclination may increase, the points n, c, r, all approach still nearer to a, and when the reclination is 90 they are all coincident, and this vanisheth into an horizontal dial whose substile will be ba. Also if the Declination be still increased, at last the points b, and m, will be coincident, and the dial plane will be parallel to the prime vertical, and the work a South dial whose substile is or may be za. mathematical diagram This, being according to the common way trodden in Trigonometry, I shall not need to prove. In like sort when any such inconveniency shall happen in South declining incliners, they that would do it without Logarithmes may work by these Analogismes. 1 Radius′. s inclination″. t declination′. t. b″. 2 s. declination′. radius″. s. b′. s. c″. Then, c + compliment of latitude taken from 180d. let the rest be called d. 3 s. c′. s. inclination″. s. d′. s. f″. 4 t. inclination′. s. b″. t. f′. s. g″. Then b, is an arch whose compliment to 90, is the distance betwixt the meridian and horizon. Also c, is an arch which being added to the compliment of latitude, and the aggregate taken from a Semicircle, the residue, namely d, is an arch composed to find f, which is an arch equal to the height of the stile, or Pole above the Plane. Lastly, g or the compliment thereof to 90 deg. is an arch equal to the distance of the substile from the meridian of the place. And these are enough for any man that is but indifferently skilled, to finish the dial with, which being deduced from M. Wells, in his Chap. 20. I shall not need to prove. This is for such planes as incline more than the distance betwixt the Zenith and the Aequator. Almicanters Are lesser Circles of the Sphere, and may be called the Parallels of declination from the horizon; having in all respects the same relation and habitude to the Azimuths, that the Signs have to the meridians, although these are accounted by 15d. and those usually by 10d. and therefore as in the description of the Signs an horizontal dial proper to the plane being first (obscurely) delineated, it was showed that the points through which the signs or Parallels must pass upon every hour, might be had by applying the tangents of the compliments of the Sun's height at those hours in those Paralles, from the foot of the perpendicular stile to the respective hours; so here making use of that Azimuth which is perpendicular to the Plane; (which in all planes is that which passeth through the foot of the perpendicular stile) the rest of the Azimuths being also inscribed, the tangents compliments of the Sun's height above the plane, when he is in any Azimuth applied from the foot of the stile to the said Azimuth gives a point, through which that height, or almicanter upon that Azimuth must pass. Example, in the Triangle pzs, let there be given the compliment of Elevation pz, the compliment of the Declination ps, and the Azimuth pzs, to find the compliment of the Sun's height zs. mathematical diagram Suppose the side sz continued till a Meridian from p, cut it at right angles in c. Then first it is tc. pz′. Radius″. s. c. pzc′ t. zc″. Secondly, s. c. pz′. s. c. ps″. s. c. zc′. s. c. sc″. So ze and se, being severally found, the difference betwixt them namely see − ze, is the compliment of the Sun's height above the Horizon. Then find how high the Sun is above the plane of the dial at the same time, the tangent compliment of that height applied, from the styles foot to the Azimuth representing the angle pzs, gives upon it the Almicanters' point, or passage. Or because s, ps′. s, pzs″. s, zs′. s, zps″. the hour from Noon, that is the angle zps, is found, which will cross the azimuth aforesaid in the same point also. Which hour if it be uneven, and unfit to remain with the rest, may be drawn obscurely. Of the Jewish, Babylonish and Italian hours. The Babylonish are accounted equal hours from Sun rising, and may be inscribed upon any Plane by help of those two parallels, which show the longest, and shortest day consisting of entire hours, as here 16 and 8 hours, and of the Aequator; for a line drawn through the hour of 5 in the first, 7 in the aequator, and 9 in the other, is the hour of 1 from Sun rising. Likewise in the same order, through 6. 8. and 10 shall pass the hour of 2, the like order in all. In Winter when the parallel of 8 hours shall fail, the other two points will serve; because the hours to be drawn are right lines. But after the first six hours are inscribed, the Aequator failing also, some other diurnal arch as of 9 or 10 hours must be described to supply that want. The Italian hours are accounted 1, 2, 3, etc. from Sun setting; to inscribe these the same diurnal arches will serve, and a line drawn through them in the hours 9 7. and 5. after noon, (observing the same order as before) shall be the hour of 1. the like through 7. 5. and 3. shall be the hour 23, the night hours of 9, 10, etc. are the morning hours produced. The Jewish hours are reckoned like the Babylonish, from Sunrising, but unequally, their sixth hour being noon; and every hour a twelfth part of the artificial day, of what length soever that be. The vulgar hours proper to the Plane being first drawn, and the Diurnal arches of 15. 12. and 9 (if it may be) divide the degrees in each by 12. and the quotients by 15. or else (which is all one) divide the said arches by 180, the three quotients shall give the just times in hours or usual parts of hours from 12 of clock upon the two Parallels and the aequator; through which lines drawn by a Ruler shall be the Jewish hours desired. Example, in latitude 51. 32′, the diurnal arch of 15 hours, is in degrees 225, which divided by 180 the quotient is 1¼. and so much the Jewish hours of 5 and 7 are distant from noon, an hour and quarter being a twelfth part of the diurnal arch of 15 hours, which hour and quarter being doubled, gives the place of 4 and 8, tripled the place of 3 and 9, etc. from noon, upon that parallel of 15 hours. In like manner the diurnal arch of 9 hours, is 135 d. which divided by 180, quotient is 135/180 that is ¾ of an hour, which shows the place of the Jewish 7 and 5, to be three quarters after or before noon, and doubled is 1½. which gives the place of 8 and 4, all one with our 1½. and 10½. and so tripling and quadrupling and quintupling of ¾ gives the places of the other hours on this parallel of 9 degrees. And these parts doubled and tripled as is said, will always (in this parallel and the former) fall upon even hours halves or quarters of our hours, which is the only reason why these two parallels of 15 and 9 are preferred; there being no necessity of using them more than the tropiques or other parallels, only this conveniency of even parts. Lastly, in the diurnal arch of 12. that is, the Aequator the equal and unequal hours concur, that is, the Jewish hours of 5 and 7, with our hours of 11 and 1, so their 4 and 8, with our 2 and 10, etc. so that a line drawn from 1¼. in the arch of 15 to 1, in the aequator, and from thence to ¾ in the arch of 9 is their 7, etc. The Circles of Position I omit, not for that the inscription of them in any plane is difficult, but where the labour is not much, and the use of the thing not any, I hold that labour too much. The way to describe dials upon Rings, Quadrants and Cilinders, as also Globes, and Concave-Hemispheres, I also pass over: not for the same reason as the other; for of all these there is much use, and pleasure in using; but because every man that shall have travailed through dialling, on planes, with the dresses thereto belonging, cannot possibly want so much ingeniosity, as may direct him to do these without book. CHAP. XVIII. IT remains only to say something of the form of the Parallels upon Planes; which (when they are not circles, (as under the Poles) are all and always Conique Sections, as is showed by Mydorgius lib. 4. Prop. 34. And here shall be showed how at any time or place it may be known what Section it is; although this is not necessary, for the Dialler to know; because without knowing them he may draw them upon the plane, as hath been showed already. Take therefore these three brief Rules, which by Aguilonius lib. 6. Prop. 83. are proved. Rule. 1. When the Sun is in any Parallel, if the plane of the dial be parallel, to a great Circle of the Sphere which toucheth the parallel, and the opposite thereto, the projection of the shadow is a Parabola. Rule. 2. If the Dial plane be equidistant, to a great Circle which cuts the parallel, and the opposite, the shadow runs in an Hyperbola. Rule 3 If the dial plane be equidistant, to a great Circle which neither toucheth, nor cutteth the Parallel, the shadow (of the perpendicular stile, for so we mean all this while) gives an Ellipsis. Example. Let the Poles elevation be (in the Scheme follow,) the arch ep. The Horizon we. The Meridian wnes. The Aequator aet. cl the Ecliptic. cy, xl the Tropiques. bd any other Parallel. n, s, Zenith and Nadir. Now therefore it is clear, that the horizontal dial in latitude 51. 32. being equidistant to the great circle we, which cuts the tropiques, and all the parallels between them, as bd, or any other, is (according to that which hath been said) such, as that the shadow thereupon all the year long shall describe Hyperbolaes'; but of different kinds, as it shall cut several parallels more or less unequally. mathematical diagram But if a dial were made parallel to the Ecliptic cl, which toucheth the Tropiques, the shadows thereon when the Sun is in the Tropiques would be Parabola's. Lastly, if the plane were equidistant, not to the aequator aet, but to some other plane mk, whose great circle neither touches any of the parallels, nor cuts them, the shadow there shall always trace some Ellipsis; not always the same, but lesser, as the Sun, or his parallel approacheth towards the aequator: but greater in those horizons which make more acute angles with the aequator; until at last the horizon and the aequator being coincident, the projecture of the shadow shall be a Circle. Likewise the horizon howsoever situate, if the Sun be in no Parallel, the Projecture is a right line. It shall not need to bring hither the demonstrations, which Aguilonius useth to prove all this; for the whole matter with small ado may be manifested thus. The Sun being in the southern signs, suppose the dark Cone cay, in North latitude to be cut by a plane cay, through the Vertex a, perpendicular to the centre of the base, it gives the triangle cay, for the flat and under superficies of the semicone cay. And let ro, be the horizon, or dial plane, (for every dial plane is parallel to some horizon) and let it be equidistant to ca; then the common section ro, (which while the Sun is in the tropic x, distinguisheth, or rather separateth the light from the darkness) is by the Definition general, of Chap. 11. A Semiparabola, in like sort might be proved, if the Sun were in any other parallel, as suppose bd, for supposing the pricked line ro, to represent the dial plane, parallel to ba, the same conclusion follows by the said Definition. Secondly, let the dial plane be R ☉, parallel to the great circle we, which cuts the parallel cy, in ☉, then by the said Definition general, the common section R ☉, which that day separates the light and darkness upon the Plane, is a semi-hyperbola: or if instead of the semicone, one conceive the section of the Cone through the vertex and axis, to be the plane triangle cay, and R ☉ as a right line, to be only the diameter of the section, the thing is the same, and the section by the second of the third definitions of Mydorgius, an Hyperbola. Lastly, (to avoid confusion of lines) let the Sun be in the northern signs, and xal the dark Cone, and zl the dial plane in South latitude equidistant to the great circle mk, which neither toucheth nor cutteth the parallels, it is evident that zl, or any line equidistant to mk, shall cut the triangle xal, made by a section from the vertex, as before, in both the sides xa, and lafoy, and is therefore by the third of the third Definitions of Mygorgius the diameter of an Ellipsis, or conceiving all by the general definition of Chap. 11. as before, it is half an Ellipsis. And so any other line parallel to zl, and greater than xl, is the diameter of a greater Ellipsis, or rather an Ellipsis of a greater Cone, which might be made by producing the lines axe, and all at pleasure. And this is proof enough, for all that hath been said of this matter; and enough hath been said (I hope) to make it intelligible. I will now show how the Sun's height for every hour of the day, in the beginning of any sign, within any latitude may be found. Let the Latitude of the Place, or the dial plane, be 19 25′. and the sign the beginning of Taurus, declining 11. 30. as in Chap. 17. And let it be required to find the Sun's height, at any hour that day in that place. mathematical diagram The easiest way to work, is by Logarithmes of the sins and tangents, because so Addition and Subtraction supplies for Multiplication and Division. The side pz, is the compliment of the latitude, and is 70. 35′. likewise the side ps, is the compliment of the declination, and is 78. 30′. and the side sz, is the compliment of the Sun's height, or the compliment of the thing required. Now then suppose the perpendicular zr, to cut ps in r at right angles; also suppose the hour required to be 4 or 8 of clock, and 4 an afternoon hour, then zps = 60. which is given, to find sz, or rather the compliment of sz. In the right angled triangle zrp, choose the angle zpr, for the middle part. Then it will be s.c. 60 + radius, whose log. are 196989700 Equal t.c. pz + t. pr subst. t.c. pz log. 95471377 Remains the tan. of pr = t. 54. 49. log. 101518323 Take 54. 49′. from 78. 30′. the rest 23. 41′. is equal to sr. But it is known in trigonometry. That s. c. rp′. s. c. rs″. s. c. pz′. s. c. sz″. Add the two middle most. That is, to s. c. 23. 41′. 99617909 Add s. c. 70. 35′. 95217074 Sum is = 194834983 From which take s. c. rp (54. 49′.) 97605692 Remaines s. c. sz = 31. 53. 97229291 Which is the height of the Sun above that Horizon, at 4 after noon or 8 before noon. And so at two operations, may any altitude, for any hour given, above this, or any other horizon be certainly found. If any like the natural sins and tangents better, he hath three things given to find a fourth, for first. t. c, pz′. s. c. rpz″. Radius′ t. pr″. Secondly s. c. pr′. s. c. sr″. s. c. pz′. sc. sz″. as before. The perpendicular zr, shall ever (for the hour betwixt 6 and noon) fall within the triangles, because the angles zsp, and zps are both acute, as in Note 2. of Definition 2. At 6 of clock the angle zps, is 90d. and s. c. zs + r = s. c. pz + s. c. ps. I put r, for Radius, always for abridgement. At the hours after 6 at evening, or before in morning the said angle zps, is obtuse; then the same triangle remaining. It is, s. c. pr′. s. c. pr + ps″. s. c. zp′. s. c. zs″. The arch pr, being first found, as before. Or it is the same in hours equidistant from 6. A Table of semidiurnal arches, in the beginning of every sign for 32 divers Latitudes. Po. alt ♋ ♊ ♌ ♈ ♎ ♓ ♍ ♒ ♐ ♑ D. d. ′ d. ′ d. ′ d. d. ′ d. ′ d. ′ 35 107 44 104 56 98 11 90 81 49 75 4 72 16 36 108 25 105 30 98 30 90 81 30 74 30 71 35 37 109 8 106 6 98 49 90 81 11 73 54 70 52 38 109 52 106 42 99 9 90 80 51 73 18 70 8 39 110 37 107 20 99 29 90 80 31 72 40 69 23 40 111 24 107 59 99 50 90 80 10 72 1 68 38 41 112 12 108 39 100 11 90 79 49 71 21 67 48 42 113 3 109 21 100 33 90 79 27 70 39 66 57 43 113 55 110 4 100 56 90 79 4 69 56 66 5 44 114 50 110 49 101 24 90 78 40 69 11 65 10 45 115 46 111 35 101 44 90 78 16 68 25 64 14 46 116 46 112 24 102 10 90 77 50 67 36 63 14 47 117 48 113 14 102 36 90 77 20 66 46 62 12 48 118 53 114 7 103 4 90 76 56 65 53 61 7 49 120 1 115 2 103 32 90 76 28 64 58 59 59 50 121 13 116 0 104 2 90 75 58 64 0 58 47 51 122 29 117 1 104 33 90 75 27 62 59 57 31 52 123 49 118 6 105 6 90 74 54 61 54 56 11 53 125 15 119 14 105 40 90 74 20 60 46 54 45 54 126 46 120 26 106 16 90 73 44 59 34 53 14 55 128 23 121 42 106 53 90 73 7 58 18 51 37 56 130 8 123 3 107 33 90 72 27 56 57 49 52 57 132 2 124 31 108 15 90 71 45 55 29 47 58 58 134 6 126 4 109 0 90 70 00 53 56 45 54 59 136 21 127 46 109 47 90 70 13 52 14 43 39 60 138 52 129 35 110 38 90 69 22 50 25 41 8 61 141 40 131 39 111 32 90 68 18 48 25 38 20 62 144 52 133 47 112 30 90 67 30 46 13 35 8 63 148 35 136 13 113 32 90 66 28 43 47 31 25 64 153 3 138 58 114 39 90 65 21 41 2 26 57 65 158 49 142 6 115 52 90 64 8 37 54 21 11 66 167 35 145 44 117 11 90 62 49 34 16 12 25 A Table of the amplitude in the beginning of every sign, for 27. Elevations of the Pole. Eleva. of the Pole ♋ ♊ ♌ ♉ ♍ ♈ ♎ In these 2 signs there is no ampl. D. d. ′ d. ′ d. ′ 36 29 29 25 13 14 15 In the other opposite signs ♏, ♓; ♐, ♒ and ♑, the same Table may serve, giving to opposites equal amplitude as to ♑, in each degree the same as to ♋, to ♐ and ♒, the same as to ♊ and ♌, lastly to ♏ and ♓, the same as to ♍, and ♉. 37 29 55 25 34 14 26 38 30 21 25 57 14 38 39 30 49 26 20 14 51 40 31 19 26 48 15 4 41 31 51 27 11 15 18 42 32 24 27 38 15 32 43 32 59 28 7 15 48 44 33 37 28 38 16 4 45 34 16 29 11 16 21 46 34 59 29 45 16 39 47 35 43 30 22 16 58 48 36 31 31 1 17 19 49 37 22 31 42 17 40 50 38 17 32 26 18 3 51 39 15 33 13 18 27 52 40 18 34 3 18 52 53 41 26 34 57 19 19 54 42 39 35 55 19 48 55 43 58 36 57 20 19 56 45 24 38 4 20 51 57 46 59 39 16 21 26 58 48 43 40 35 22 4 59 50 38 42 1 22 44 60 52 47 43 35 23 28 61 55 13 45 20 24 15 62 58 1 47 15 25 5 A Table of arches of the Horizon intercepted betwixt the Meridian, and each hour line upon the Dial Plane, for horizontal and Vertical Dial's, Calculated for 21 Elevations of the Pole. The Altitude of the Pole for Horizontals. 12 11.1. 10.2. 9.3. 8.4. 7.5. 6. The Altitude of the Pole in Verticals. d. ′ d. ′ d. ′ d. ′ d. ′ d. ′ d. 35 8 43 18 18 29 49 44 49 64 35 90 55 36 8 57 18 46 30 32 45 30 65 29 54 37 9 10 19 9 31 2 46 11 66 0 53 38 9 22 19 34 31 37 46 50 66 29 52 39 9 33 19 58 32 11 47 28 66 55 51 40 9 45 20 21 32 44 48 7 67 21 50 41 9 57 20 44 33 16 48 39 67 47 49 42 10 10 21 7 33 46 49 12 68 11 48 43 10 22 21 29 34 18 49 44 68 33 47 44 10 32 21 51 34 47 50 16 68 54 46 45 10 43 22 12 35 17 50 46 69 15 45 46 10 54 22 33 35 44 51 15 69 35 44 47 11 5 22 53 36 11 51 42 69 53 43 48 11 17 23 13 36 37 52 9 70 11 42 49 11 25 23 33 37 3 52 35 70 28 41 50 11 35 23 52 37 28 53 0 70 43 40 51 11 45 24 9 37 52 53 24 70 59 39 52 11 55 24 27 38 15 53 46 71 13 38 53 12 5 24 43 38 37 54 8 71 28 37 54 12 13 25 2 38 58 54 29 71 41 36 55 12 22 25 18 39 19 54 49 71 54 35 These Tables need no explanation, the use of them being evident. But if they prove not satisfactory, for want of calculation, for further degrees of Elevation; or for want of halves, and quarters of degrees, or the like of hours, they are as I had them out of Kercherus his Ars Magna. Nevertheless I will show the making of them, whereby any man may fit them for his own purpose, and for his place, (if it happen without these limits) by his own calculation, as followeth. First for the Table A. Radius′ t. c. Elevation″. t. c′. Declination′. b″. This b, is the sine compliment of the angle at the Pole, which shows the hour from Noon, in Winter; and the hour from Midnight in Summer, wherein the Sun riseth, having declination, which declination is ready in tables, the making of which shall be showed anon and also the table of the Sun's declination at the end hereof shall ensue. The angle at the Pole, so found being divided by 15. shows in Winter the semidiurnal arch in hours, which was had by the first working in degrees and minutes. And in Summer the seminocturnal; whose compliment to 180 degrees or to 12 hours, is the thing required, here all the time from the vernal to the Autumnal Aequinoctial, is called Summer. Secondly, for the Table B. s. c. Elevation′, s. Declination″, Radius′. s. c. Azimuth″, which Azimuth being compared with 90 d. difference is the Amplitude. Example, for Elevation 40. initio ♊ To Radius log. 10000000 Add Sine declination 20. 13′. log. 09538537 Sum 19538537 Subtract sine compliment elevation, Remains Sine compliment Azimuth, which compliment here is the Amplitude 9654283 The arch belonging to sine 9654283. being sought in the Canon, is 26. 48′. which is the Amplitude required, where the Elevation is 40 degrees. Thirdly, for the Table C. First, to find any horizontal arch for any hour, as for Example 3, or 9 Radius′. t. of the hour in degrees, that is here t. 45″. Sine Elevation′. t. of the arch required″. Or else, t. c. 45′. r″. s. Elevation′. t. arch required″. Secondly, in a Prime Vertical. Radius′. t. hour″. s. c. Elevation′. t. of the arch″. It is at first in the Symbols, Chap. 1. advertised that the letters s. and t. or s. c. and t c. signify the sine, and tangent; or sine compliment and tangent compliment of an arch or angle. And working by the Logarithmes of the sins and tangents, the former Analogismes happen not but in their stead certain Aequalities, or Aequations, as follow. For the Table A. tc. Elevation + tc. declination = Rad: + b. Which b, is the co-sine of the thing required, that is, of the angle at the Pole, which divided by 15, giveth the time. For the Table B. Rad: + s. declination = sc. elevation + sc. Azimuth. For the Table C. s. Elevation + t. the hour = Radius + t. the arch, or more readily thus, tc. hour′ r″ s. Elevat′ t. of arch from the substile″. It must still be remembered that r, stands for Radius. The Elevation is always taken for the height of the Pole above the horizon, which horizon is the dial plane. In other planes, as the Prime Vertical, and all other verticals, the height of the Pole above the plane must be used, having therefore found that, call it p, or else call the declination q, then s. q + s. elevat. = r + s. p. And, r + s, p, = tc. the hour + t, the arch of that hours distance from the substile. So after still till all the hours be found, this later work must be repeated. Whereas we use tc. the hour, and t. the arch, by the hour is always meant the angle at the Pole, or the space there included between any hour, and the substile; as 15d. for 1, 30d. for 2, etc. The arch is the distance of any hour from the substile measured in the arch of a circle, whose centre is the centre of the dial, when it is projected upon the plane. To find the Declination of a Place. The declination is an arch of a great circle passing through the Poles of the World, and the centre of the place whose declination is sought, intercepted between the said centre and the Aequator. If the place have no latitude, that is, if it be in the Ecliptic, the nearest distance from Aries or Libra being given, call it b. Then r′ s. 23d. 32 m″ s. b′ s. e″ and e, is the declination required: working by the Natural Sins. Secondly, if the place have latitude, that being given, or found in Tables, and the right angle which the circle of latitude makes with the Ecliptic, (for all circles of latitude do so, as the circles of declination do with the Aequator) and the next distance to Aries or Libra being also given. 1 Then if the place lie betwixt the Ecliptic and the aequator, call the nearest distance to ♈ or ♎, b, as before. And the latitude given c. It is Logarichmically s. c + r = s, b + s, a & a, is an angle, which being taken out of 23. 32′. leaves an angle, which angle call d, then s, b + s, d = r + s, e. And e, is the declination required. 2 If the place lie betwixt the Ecliptic and the Pole, the angle a, found as before, must be added to 23. 32′. and call the sum f, then s, b + s, f = r + s, e, etc. 3 Lastly, let the place lie betwixt the aequator and the other Pole, then s, c + r = s, b + s, a, and from a, subtracting 23. 32′. call the rest g. Then s, b + s, g = r + s, e, and e, the declination. To find the right Ascension of a Place. If it be in the Ecliptic, as the Sun is, let the nearest distance from Aries be called still b. And working by Logarithms, it is r + sc, 23. 32′. = = tc, b + t, a. And a is the right Ascension. 2 If the place have latitude, call it still c, and let the declination found with latitude, as before, be called, q. Then sc, b, + r = sc, q + sc, a and a, is the right ascension; or (between ♋ and ♎) the compliment of it. To find the Ascensional Difference. Thus t. elevation + t. declination = r + s.y. And y, is the Ascensional difference. To find the, Obliqne Ascension. In the southern signs add the Ascensional difference to the right Ascension: or in the northern signs subtract the same from the right Ascension, the sum in Winter, and the remainder in Summer, is the obliqne Ascension. That which hath been last said concerning the Declination, Right Ascension, Ascensional diffeence and obliqne Ascension, may be illustrated, and demonstrated also out of this figure, at least with some small variationr. mathematical diagram In which let ho, be the horizon, aeq, the aequator, pp the Axis of the World, ec the Ecliptic, mn, the parallel in which the Sun is at d. po the Elevation, z the points ♈ and ♎, bz. the nearest distance from one of those points. Which is supposed given, or known, dr the declination. zr the right Ascension of the Sun: or sometimes the compliment of it. The angle xpo, is the compliment of the Ascensional difference, yq the measure of it. Therefore here in Winter ry, is the obliqne ascension, but the Sun being now supposed to be in d, that is 20 degrees of Taurus, by that which hath been said before, rz − zy, is the obliqne ascension. All these may be found (enough being given) as first in the triangle drz right angled at r, are given the side dz = 5 cd. the angle dzr = 23. 32′. and the angle r 90 d. to find the declination dr. Working by the Artificial sins and tangents, put br, for the middle part, therefore, s. dr + radius, = s. dz + s. dzr, which resolved into proportionalty by the 14. of the 6. of Euclid, will be. Radius′ s. dz″ s. dzr′ s. dr″ or, Alternately, Radius′ s. dzr″ s. dz′ s. dr″, the thing sought. And the Analogisme the very same with that shown before for finding the declination. Then for the right ascension, zr make dzr the middle part, it is sc. dzr + Radius = tc. dz + t. zr. And resolved, tc. dz′ Rad″: sc. dzr′ t. zr″. That is Radius′ t. dz″ sc. dzr′ t. zr″. which is the same Analogisme with that before, for finding the right ascension. Secondly for the ascensional difference in the triangle xpo, right angled at o, put the angle xpo, for the middle part. Then sc. xpo + Radius = t. po + tc. xp but tc. xp = t. zy. That is, Rad′ t. po″ t. xy. (or br′) sc. xpo″ And the compliment of xpo, is xpz. Whose measure is zy, the ascensional difference sought for. Example of all, first for the declination rd, put rd for the middle part. Unto, s. dz, 50. log. 9884254 Add, s. dzr, 2 3 ½. log. 9601570 Sum is 19485824 And subtracting Radius remains. Which is the sine of 17. 49′. for the declination. Then for the right Ascension rz, put that for the middle part, and Unto, t. dr, 17.49. log. 9507027 Add, tc. dzr, 66.28 log. 10361007 Sum 19868034 Whence subtracting Radius, remain. 9868034 The sine of 47. 33′, for rz: the right ascension. Secondly, for the ascensional difference zpx, in the triangle xpo, putting the angle xpo, for the middle part, Unto t. po, 51. 32′. log. 10099913 Add, tc. px, that is t. br, 17. 49′. log. 09507027 Sum is 19606940 From which taking Radius, rests 09606940 The Sine compliment of xpo, or the sine of xpz = 23. 51′. the Ascensional difference. Which being taken from rz, the right ascension 47 33. remains rz − zy = 23.42. for the obliqne ascension. Note 1. The compliment of the ascensional difference is equal to the quantity of hours and parts betwixt midnight and sunrising. NOTE. 2. The obliqne ascension of the Sun being taken, from the right ascension in Summer, the residue is equal to the excess, whereby the semidiurnal arch is more than 6 hours: also the right ascension taken from the obliqne ascension in Winter, the rest is the defect whereby the semidiurnal arch is less than 6 hours. And the right and obliqne ascension are nearer to equality, as the Sun attaineth near either ♈ or ♎. If the place be not in the Ecliptic, but hath latitude, as the asterisme ⚹ in the last figure, if there be given that latitude d ⚹, and the distance from the next aequinoctial point z ⚹, these with the right angle at d, are sufficient to find out all, in manner as hath been showed before. HEre follow two Tables of Interest, whereof the first showeth how much 100 li. with all its increase by means of Compound Interest at the several rates of 5, 6, 7, 8, 9, and 10 per. Cent. amounts to annually for 31 years: The second showeth the like increase for 100 li. Annuity or yearly rent, at the like rates, and for the same term. In both which the first Column towards the left hand shows the number of years successively to 31, the second gives the increase together with the Principal in entire pounds sterlin; the third hath the Numerators of the fractions of a pound to be added to their respective Integers. Only in the rate for 6 per. Cent. (which is now of more frequent use) the fractions are reduced to shillings and pence, (ommitting less than pence) as may easily be seen by the table. And the fractions in the other rates (whose common denominator is 1000000) may be easily enough, either so reduced, or very nearly guessed, by such as are but moderately versed in Arithmetic. ye. At 5 pe. Cent. ye. At 6 p. Ce. 1 105 000000 1 106 2 110 250000 2 112 7 2 3 115 762500 3 119 2 0 4 121 550625 4 126 4 11 5 127 628156 5 133 16 5 6 134 009563 6 141 17 0 7 140 710041 7 150 7 3 8 147 745543 8 159 7 8 9 155 132820 9 168 18 11 10 162 889461 10 179 1 8 11 171 033934 11 189 16 7 12 179 585630 12 201 4 4 13 188 564911 13 213 5 10 14 197 993156 14 226 1 9 15 207 892813 15 239 13 1 16 218 287453 16 254 0 8 17 229 201825 17 269 5 6 18 240 661915 18 285 8 8 19 252 695062 19 302 11 2 20 265 329845 20 320 14 2 21 278 596337 21 339 18 9 22 292 526154 22 360 7 0 23 307 152461 23 381 19 5 24 322 510084 24 404 17 10 25 338 635588 25 429 3 11 26 355 567367 26 454 18 11 27 373 345735 27 482 4 10 28 392 013021 28 511 3 7 29 411 613672 29 541 17 0 30 432 194355 30 574 7 2 31 453 804072 31 608 16 5 ye. At 7 pe. Cent. ye. At 8 p. Cent. 1 107 000000 1 108 000000 2 114 490000 2 116 640000 3 122 504300 3 125 971200 4 131 079601 4 136 048894 5 140 255173 5 146 932807 6 150 073035 6 158 687431 7 160 578147 7 171 382355 8 171 818617 8 185 092943 9 183 845920 9 199 900378 10 196 715134 10 215 892408 11 210 485193 11 233 163800 12 225 219156 12 251 816904 13 240 984497 13 271 962256 14 257 853412 14 293 719236 15 275 903151 15 317 216774 16 295 216371 16 342 594116 17 315 881519 17 370 001645 18 337 993225 18 399 601776 19 361 652750 19 431 569818 20 386 968442 20 466 095403 21 414 056231 21 503 383035 22 443 040167 22 543 653677 23 474 052979 23 587 145971 24 507 236687 24 634 117648 25 542 744255 25 684 847059 26 580 736353 26 739 634823 27 621 373898 27 798 805608 28 664 870071 28 862 710056 29 711 410976 29 931 726860 30 761 209744 30 1006 265009 31 814 494426 31 1086 766210 ye. At 9 p. Cent. ye. At 10 p. Cent. 1 109 000000 1 110 000000 2 118 810000 2 121 000000 3 129 502900 3 133 100000 4 141 158161 4 146 410000 5 153 862395 5 161 051000 6 167 710010 6 177 156100 7 182 803911 7 194 871710 8 199 256283 8 214 358881 9 217 189348 9 235 794769 10 236 736389 10 259 374246 11 258 042664 11 285 311671 12 281 266504 12 313 842838 13 306 580489 13 345 227122 14 334 172733 14 379 749834 15 364 248279 15 417 724817 16 397 030624 16 459 497299 17 432 763380 17 505 447029 18 471 712084 18 555 991732 19 514 166173 19 611 590905 20 560 441127 20 672 749995 21 610 880828 21 740 024994 22 665 860103 22 814 027493 23 725 787512 23 895 430242 24 791 108388 24 984 973266 25 862 308143 25 1083 47053 26 939 915876 26 1191 817652 27 1024 508305 27 1310 999417 28 1116 714052 28 1442 099359 29 1217 218317 29 1586 309295 30 1326 767966 30 1744 940225 31 1446 177082 31 1919 424247 The second Table for 100 li. year rent, or Annuity. ye. At 5 p. Cent. ye. At 6 p. Cent. 1 100 000000 1 100 2 205 000000 2 206 3 315 250000 3 318 7 2 4 431 012500 4 437 8 7 5 552 675625 5 563 13 6 6 680 309406 6 697 9 11 7 814 324876 7 839 6 11 8 955 041119 8 989 14 2 9 1102 793174 9 1149 1 11 10 1257 932832 10 1318 0 10 11 1420 829473 11 1497 2 6 12 1591. 870946 12 1686 19 0 13 1771 464493 13 1888 3 4 14 1960 037717 14 2101 9 2 15 2158 039602 15 2327 10 11 16 2365 941582 16 2567 4 0 17 2484 238661 17 2821 4 7 18 2813 450594 18 3090 10 1 19 3054 123123 19 3375 18 8 20 3306 829279 20 3678 9 10 21 3572 170742 21 3999 4 1 22 3850 779279 22 4339 3 1 23 4142 318242 23 4699 10 0 24 4449 434154 24 5081 9 5 25 4771 905861 25 5486 7 2 26 5110 501154 26 5915 10 10 27 5466 026211 27 6370 9 6 28 5839 327521 28 6852 14 0 29 6231 293897 29 7363 17 3 30 6642 858591 30 7905 14 9 31 7075 001520 31 8480 0 9 ye. At 7 p. Cent. ye. At 8 p. Cent. 1 100 1 100 2 207 2 208 3 321 490000 3 324 640000 4 443 994300 4 449 611200 5 575 073901 5 585 580096 6 715 329074 6 732 426503 7 865 401609 7 891 019623 8 1025 979721 8 1062 301192 9 1197 798301 9 1247 285287 10 1381 644182 10 1446 068109 11 1578 359274 11 1661. 753557 12 1788 844423 12 1893 693941 13 2014 063532 13 2145 189456 14 2255 047979 14 2416 804612 15 2512 901337 15 2710 148980 16 2788 804430 16 3026 960898 17 3084 020740 17 3369 117769 18 3399 902191 18 3738 647190 19 3737 895344 19 4137 738965 20 4099 548018 20 4568 758082 21 4486 516379 21 5034 258738 22 4900 572525 22 5536 999437 23 5343 612601 23 6079 959391 24 5817 665483 24 6666 356142 25 6324 902066 25 7299 664633 26 6867 645210 26 7983 637803 27 7448 380374 27 8722 328827 28 8069 767000 28 9520 115133 29 8734 650690 29 10381 724346 30 9446 076238 30 11312 262290 31 10207 301574 31 12317 243213 ye. At 9 p. Cent. ye. At 10 p. Cent. 1 100 1 100 2 209 2 210 3 327 810000 3 331 4 457 312900 4 462 100000 5 598 471016 5 608 310000 6 752 333456 6 769 141000 7 920 043467 7 946 055100 8 1102 847379 8 1140 660600 9 1302 103643 9 1354 726600 10 1509 292970 10 1590. 199200 11 1745 129337 11 1849 219100 12 1992 190977 12 2134 141000 13 2271 488165 13 2447 575100 14 2575 922099 14 2792 310610 15 2907 755087 15 3171 541671 16 3209 453045 16 3588 695838 17 3598 303819 17 4047 565421 18 4022 150662 18 4552 321925 19 4484 144223 16 5107 554159 20 4987 717203 20 5718 309575 21 5536 611751 21 6390 140532 22 6134 906809 22 7129 154585 23 6787 048420 23 7942 070043 24 7497 882777 24 8836 277047 25 8272 692226 25 9819 904751 26 9117 234526 26 10901 895226 27 9932 186348 27 12092 084748 28 10932 186348 28 13401 293222 29 12016 083110 29 14841 422544 30 13197 530589 30 16425 564798 31 14485 308342 31 18168 121277 The use of these Tables is thus. If it be asked how much 100 li. put forth to use comes to in 17 years, at 6 per cent. compound interest? Look for the title of 6 per cent. at the top of the leaf, and in the first Table, than also look down in the column entitled Years, till you find the number 17, just over against 17, towards the right hand you shall find 269 li. 5 s. 6 d. which is the thing required. Or if it be asked how much 100 li. per Annum in rend amounts to in 13 years at 8 per cent. compound Interest? Look in the second Table for 13 years, and under 8 per cent. you shall see 2145 li. and 189456/1000000 of a pound, that is reduced, 2145 li. 3 s. 9 d. and something more, which more, being less than a penny, I omit, as (in this case) not considerable. NOTE. Although I say it amounts to so much, yet I do not say it is worth so much; for who would part with 2145 li. 3 s. 9 d. presently, in hope to get it up again in 13 years by 100 li. per annum? When money was at 8 per cent. a Lease of 21 years was accounted by some worth 9½ years purchase, by others worth 10 years' purchase: so that 100 li. per annum rend at the most, is worth but 1000 li. in 21 years: that is (by the rule of three) 619 li. 11 d. for 13 years. Note 2. It is also fit to be known, that proportionally as Money is less valued, land is more, et contrà. So that according to 10 years, purchase for Rend Charges or Annuities for years, when Money was at 8. per. Cent. the money being now at 6. per. Cent. the purchase must be 13 years 4 months rend of the land, etc. So likewise, Money being at 8. per. Cent. land for ever used to be sold for 20 years' Rent, but now (if no external accident hinder) it ought to be sold for 26⅔. times the yearly rent thereof. For 6′ 8″. 20′ 26⅔. Also, 6′ 8″. 10′ 13⅔″. And the like Analogisme will serve being used in other rates. As if money were at 5 or 7. per. Cent. then 5′ 8″. 20′ 32″. or 7′ 8″. 20′ 22 6/7″. Also 5′ 8″. 10′ 16″. Or 7′ 8″. 10′ 11 6/14″. And so of any. This, considering the largeness and clearness of the Tables, is all I mean to say concerning that compound interest which is called Direct or Profitable. There is another sort of Interst which gives the yearly decrease of 100 li. or any other Sum. And this is called Compound Interest Rebated, or Damageable. Which Decrease is orderly made by subtracting the Interest from the principal yearly, as the increase of 100 li. in the former Tables was caused by adding the yearly interest. Example. If 100 give 6, then at the end of the first year the 100 li. is increased, and become 106, and so again, if 100 give 6, what 106? it makes 6 36/100 which add to 106, than the said 100 li. is at the end of two years hereby increased to 112 36/100, and so the first Table is made for every year. But now if 100 give 6, what shall 94 give? it is 5 84/100, which taken from 94, rests 88 16/100, so the 100 li. at the end of the first year is decreased to 94 li. and at the end of the second to 88 16/100. But because the composing Tables for this is much labour as that which hath been done already, also for variety's sake, I will add a third Table (which I take out of Simon Stevens Practical Arithmetic) consisting of artificial numbers, which will serve as well for direct as rebating interest. And when I have showed the way to make and use those Tables, and put a few Problems requisite and difficult, and adjoined a Table of the Sun's Declination, I mean to conclude this Treatise. Here followeth the third Table. The Table for 5. p. C. for 6. per. Cent. for 7. ptr. Cent. The Table for 8 p. C. for 9 per. Cent. for 10. p. Cent. 1 9523810 9433962 9345794 9259259 9174312 9090909 2 9070295 8899964 8734387 8573386 8416800 8264463 3 8638376 8396192 8162979 7938322 7721835 7513148 4 8227025 7920936 7628952 7350298 7084252 6830135 5 7835262 7472581 7129862 6805831 6499314 6209214 6 7462154 7049605 6663422 6301695 5962673 5644740 7 7106813 6650571 6227497 5834903 5470342 5131582 8 6768393 6274124 5820091 5402688 5018662 4665075 9 6446089 5918985 5439337 5002489 4604277 4240977 10 6139132 5583948 5083493 4631934 4224107 3855434 11 5846792 5267875 4750928 4288828 3875328 3504940 12 5568373 4969893 4440120 3971137 3555347 3186309 13 5303212 4688390 4149645 3676979 3261786 2896645 14 5050678 4423009 3878173 3404610 2992464 2633314 15 4810170 4172650 3624461 3152417 2745380 2393922 16 4581114 3936462 3387347 2918905 2518697 2176293 17 4362966 3713643 3165745 2702690 2310731 1978448 18 4155206 3503437 2958640 2502491 2119937 1798589 19 3957339 3305129 2765084 2317121 1944896 1635081 20 3768894 3118046 2584191 2145482 1784308 1486837 21 3589423 2941553 2415132 1986557 1636980 1351306 22 3418498 2775050 2257133 1839405 1501817 1228460 23 3255712 2617972 2109470 1703153 1377814 1116782 24 3100678 2469785 1971467 1576994 1264050 1015256 25 2953027 2329986 1842493 1460180 1159679 0922960 26 2812407 2198100 1721956 1352019 1063926 0839055 27 2678483 2073679 1609305 1251869 0976079 0762777 28 2550936 1956301 1504023 1159138 0895485 0693434 29 2429463 1845567 1405629 1073276 0821546 0630395 30 2313774 1741101 1313672 0993774 0753712 0573086 The construction of these Tables, or any other the like is as followeth. Having made choice of some great decimal number, I mean, so it may consist of all cyphers, except unity towards the left hand, as in these it is 10000000 (which shall be called the Radius of the Tables) this Radius being multiplied by 100 (which is the principal) and divided by the principal plum the Interest, the quotient is the number in the Table for the first year, which quotient being again multiplied by 100, and the product divided by principal plus interest as before, the quotient shall be the number in the Tables for the second year, and so may every years respective number be found, as was the second. Example in Interest 7 per cent. First, 10000000 into 100 gives 1000000000 which divided by 100 + 7, that is, by 107, the quotient is 9345794, which is the number answering to the first year in the Table of 7 per cent. For although there be a remain, after the division, of 42, yet because 42/107 < ½ it is here neglected: But if the remain had happened 51/107 > ½, than 1 being added to the quotient, it is 9345795, and is so much nearer the thing required. Secondly, 9345794 into 100 gives 934579400 which divided still by 107. quotient is 8734386, and 98 remaining, but 98/107 > ½, therefore adding 1 to the quotient, it is 8734387, for the number answering to the second year in the Table of 7 per cent. After this manner are all the Tables made. Use of the Tables. This shall be showed in a few Examples. Example 1. Interest profitable. If 100 li. give 6 li. for one year, what shall 500 li. give for 17 years, principal and Interest? The Rule. Multiply the Principal by the Radius, the Product is 5000000000, which divide by 3713643 (which is the number answering to 17 years in the Table of 6 per cent.) the quotient will be 1346 1444522/3713643 of Pounds. Which is the just sum of 500 l. with all its compound interest, at 6 per cent. for 17 years, and reduced it is 1346 li. 7 s. 8½ d. the like way of working will effect any question of this nature, which exceeds not the Tables in time or rates. NOTE. It may here be noted, that if 1346 li. 7 s. 8 ½ d. were due to be received 17 years hence, it is, or may be called equivalent to the receiving of 500 li. in hand, that is, such a reversion is worth 500 li. in ready money. And therefore by inversion of the former Rule, may the Rebatement, or Interest damageable be found. Example 2. Interest damageable. If there shall be 1000 li. due at the end of 21 years, and money run at 8 per cent. to be accordingly rebated, how much is this worth in ready money? The Rule. In the Table for 8 per cent. find the number answering to 21 years, which is 1986557, multiply this by the principal, the product is 1986557000, which divided by radius the quotient is 198 6557/10000. pounds. That is, 198 li. 13 s. 1 ½ d. which is the thing required. Probl. 1. If 1000 li. be to be paid at the end of 7 years and 500 li. more 2 years after that, what shall both these be worth, for borne till the end of 12 years? at 6 per. Cent. See (by the Rule belonging to Example 2.) First, what each of them is worth at the end of their proper terms, The first is, 665,0571000 The second 295,9492500 In all 961,0063500 Which is all that both are worth in ready money. The secondly, seek (as before) what 961, 0063500 li. ready money is worth 12 years hence, rebating interest of 6. per. Cent. It will come to about 478, 4778100 li. for the thing required. Probl. 2. If there be due in ready money 500 li. which at the end of 20 years will increase, and be 2330, 477015 li. what is the rate of the interest here? Take the fist part of the Number, (because the tables are made for 100 li.) which is 466,095403. and in the first table look for it against the number 20. it will be found in the rate of 8. per. Cent. and such is the interest. Or in the third table. Say, 2330, 477015′ 10000000″ 500′ 2145482″. Which last viz. 2145482. being looked for in the third table, will be found over against the year 20 under the title of 8. per Cent. which shows again that the rate of the interest is 8. per. Cent. And herein the Problem is not only cleared; but the use of both tables exemplified. Prob. 3. In like sort, if the interest, years, and total increase be given, to find the principal. As if one receive 1000 li. for compound interest at the rate of 10. per. Cent. for 7 years; how much was the principal sum? See in the third table in the rate of 10. per. Cent. what number answers to 7 years; it will be found 5131582. which being subtracted from 10000000 there resis 4868418. And then say, 4868418′ 5131582″ 1000′ 1054, 269428″. That is, the principal was 1054 li. 1 s. 1 d. and a little more, which we omit. And the same sum will be found if one use the first Table, where the interest of 100 li. for 7 years at 10 per cent. is 94,871710, for then 94,871710′ 100″ 1000′ 1054 li. 1 s. 1 d′. and a little more as before. Probl. 4. Or the rest being given, and the time required, As if there be 1000 li. due at the end of some years, and the Creditor instead of it takes 100 li. ready money, rebating compound interest at 8 per cent. at the end of what years was this at first payable? Say 1000′ 10000000″ 100′ 1000000″. Which fourth proportional number being (as near as may be) sought for in the third Table under 8 per cent. will be found to fall near 30 years: that is, the time here required is almost 30 years. Prob. 5. If 1000 li. be due at the end of 4 years, and the parties agree to have it paid at four yearly payments, that is 250 li. (rebating 6 per cent.) at the end of every year, how much is to be paid at each time? 1 Look in the third Table under 6 per cent. for 3 years, against that the number 8396192 stands, which last being multiplied by 250, and after the product divided by Radius, that is by 10000000, the quotient is 209,9048000 li. that is, 209 li. 18 s. 1 d. for the first yearly payment. 2 In the same Tables against 2 years is found the number 8899964 which used in all respects like that against 3 years already done, the quotient will be 222,499 1000, that is, 222 li. 9 s. 11 d. for the second. 3 And against 1 year is 9433962, which being multiplied like the two former, the quotient is 235, 8490500, or 235 li. 16 s. 10 d. for the third payment. 4 Lastly, Must be the full fourth, viz. 250 li. for that being not paid otherwise then in due time, suffers no Rebatement. Prob. 6. If there be a Reversion of a Lease or Annuity of 100 li. per Annun, and for 11 years to come at the end of 14 years, what is this worth in ready money? money being at 6 per cent. 1 Add 11 to 14, it makes 25 years, and look in the second Table for 6 per cent. against 25 years there stands 5486, 7, 2 d. likewise against 14 years is 2101, 9, 2 d. the difference is 3384, 18, 0 d. which cannot be received till the end of 25 years. Therefore in the third Table for 6 per cent. against 25 years, finding the borrowed number 2329986, multiply it (as hath been lately showed) by 3384 li. 18 s. 0 d. that is by 3384 9/10 li. the product divided by 10000000 is 789,4672624 That is reduced 789 li. 9 s. 4 d. And so much it is worth in ready money, and the Probl. is solved. All cases cannot be instanced, if the question happen to be like none of these, yet the Reader by his own judgement may (without doubt) resolve it by some of these Tables, to illustrate the use of which, was chief my end in putting and resolving the precedent demands A Table of the Sun's Declination for the year 1654. Day's Janu. Febru. Mar. April May Jun July Aug. Septen. Octob. Noven. Decen. South South South North North North North North North South South South 1 21 49 13 56 3 35 08 26 17 58 23 ●● 22 10 15 17 4 30 07 09 17 36 23 08 2 21 39 13 36 3 11 08 48 18 13 23 ●● 22 02 14 59 4 07 07 32 17 52 23 12 3 21 29 13 16 2 48 09 09 18 28 23 ●● 21 53 14 40 3 44 07 55 18 08 23 16 4 21 18 12 55 2 24 09 31 18 43 23 ●● 21 44 14 22 3 21 08 17 18 24 23 20 5 21 07 12 35 2 00 09 53 18 57 23 ●● 21 35 14 03 2 58 08 39 18 40 23 23 6 20 56 12 14 1 37 10 14 19 11 23 ●● 21 25 13 44 2 34 09 02 18 55 23 26 7 20 44 11 53 1 13 10 35 19 28 23 ●● 21 14 13 25 2 11 09 24 19 09 23 28 8 20 32 11 32 0 49 10 56 19 38 23 ●● 21 04 13 05 1 48 09 46 19 24 23 29 9 20 19 11 10 0 26 11 17 19 51 23 ●● 20 54 12 46 1 24 10 08 19 38 23 30 10 20 06 10 49 0 02 11 37 20 04 23 ●● 20 43 12 26 1 01 10 29 19 52 23 31 11 19 53 10 27 0 N22 11 58 20 16 23 ●● 20 31 12 06 0 37 10 51 20 05 23 32 12 19 39 10 05 0 46 12 18 20 28 23 ●● 20 19 11 46 0 14 11 12 20 18 23 31 13 19 25 09 43 1 09 12 38 20 40 23 ●● 20 07 11 26 0 S10 11 34 20 31 23 31 14 19 10 09 21 1 33 12 58 20 51 23 ●● 19 54 11 05 0 33 11 55 20 43 23 29 15 18 55 08 58 1 56 13 17 21 02 23 ●● 19 41 10 44 0 57 12 15 20 55 23 28 16 18 40 08 36 2 20 13 37 21 12 23 ●● 19 28 10 23 1 20 12 36 21 06 23 26 17 18 25 08 14 2 43 13 56 21 23 23 ●● 19 15 10 02 1 44 12 57 21 17 23 23 18 18 05 07 51 3 07 14 15 21 33 23 ●● 19 01 09 41 2 07 13 17 21 28 23 20 19 17 53 07 28 3 30 14 34 21 42 23 ●● 18 47 09 20 2 31 13 37 21 38 23 17 20 17 36 07 05 3 53 14 52 21 51 23 ●● 18 33 08 58 2 54 13 57 21 48 23 13 21 17 19 06 42 4 17 15 10 22 00 23 ●● 18 18 08 36 3 18 14 16 21 57 23 02 22 17 02 06 19 4 40 15 28 22 08 23 0● 18 03 08 15 3 41 14 36 22 06 23 03 23 16 45 05 56 5 03 15 46 22 16 23 0● 17 47 07 53 4 04 14 55 22 15 22 58 24 16 27 05 32 5 26 16 03 22 24 22 5● 17 32 07 31 4 28 15 14 22 23 22 52 25 16 09 05 09 5 49 16 21 22 31 22 5● 17 16 07 09 4 51 15 33 22 31 22 46 26 15 51 04 46 6 12 16 38 22 38 22 47 17 00 06 46 5 14 15 51 22 38 22 39 27 15 32 04 22 6 34 16 54 22 44 22 4● 16 43 06 24 5 38 16 09 22 45 22 32 28 15 14 03 59 6 57 17 11 22 50 22 3● 16 26 06 01 6 00 16 27 22 51 22 24 29 14 55 7 19 17 27 22 56 22 27 16 09 05 38 6 23 16 45 22 57 22 16 30 14 35 7 42 17 43 23 01 22 2● 15 52 05 16 6 46 17 02 23 03 22 07 31 14 16 8 04 23 06 15 34 04 53 17 19 21 58 Postscript. THe reason why in Chap. 15. I did not (as Des Cartes) continue the method to Sursolid Problems, is because the description of such curve lines, is not only difficult and laborious, but (as he confesseth) incommodious: And although he saith it is easy to find a thousand other sorts of ways, amongst which some might be better, yet I conceive it is easier for any man to believe that Des Cartes having found one way might be allowed to say this, then for that other men to find any one other way better and easier. And Des Cartes gives but only one Example, which is to find four Means, and omits quinquisection of angles, which might have been of some use, whereas the other is but of little. For the principal use of two Means seems to be in doubling the Cube, or in making rectangle Parallelepipedons' retaining any proportion given, whose three dimensions also shall be proportional: or the like of Solid Rhombi, or obliqne parallelepipedons, all which are bodies to be seen and handled, whereas four or more means to be so employed require bodies of four or more dimensions, of which we have yet no fancy. Besides, Galileo in his Book called Systema Mundi (the beginning almost) seems to prove that there can be but three dimensions in nature. Nevertheless, if any one happen to discover the Biquadraticall Body, he may then as well demonstrate that there can be but four. And yet more, seeing some Aequations of 6. dimensions require a circle to touch or cut a curve line in 6 points, which cannot be done but very obliquely, the method therefore here grows unuseful: and for septisection, and Aequations of 8 or more dimensions, it will be unsufficient. If for all this, any one hath a mind to sursolids, his aequation, by some rules going before, or here following, must be reduced to this form. a ⁶ − ba ⁵ + ca 4 − da ³ + faa − ga + h = o In which it behoveth that the quantity called c, be greater than the square of ½ b. In the section 6, of the first Rule of Chap. 4. it is showed how all the false roots in any aequation may be made true. Also in Rule the second of that Chap. it was showed how to free the aequation from the second term. There remains three other Rules now of some use. RULE 1. To cause the known quantity of the third term to be greater than the square of the like in the second, and also to change the false roots to true ones, without causing the true ones to become false. Increase the true Roots by a quantity greater than any of the false Roots. For sure it is possible to guess such a quantity, although the false roots be unknown. Example, in the aequation + aaaa + ba ³ − ccaa − ddda + ffff = o Put e − b = a Than it will be That is And making 5 bb + cc = gg & b 3 + 2 bcc + d 3 = h ³ And lastly, 2 b 4 + bbcc + f ⁴ − bd ³ = llll Then it is + e ⁴ − 3 bc ³ + ggee − h3e + llll = o In which aequation all the Roots are true by Sect. 5. Rule 1. Chap. 4. And secondly, it is manifest that gg, which is the known quantity of the third term, being equal to 5 bb + cc is greater than 9/4 bb, which is the square of half the known quantity of the second term. I have instanced in an Aequation of but 4 dimensions, for brevity sake, the work is true, or may be so, in those of six dimensions, which they that resolve sursolid Aequations will be put to. RULE 2. If yet any term of the Aequation be wanting, Increase the Root e never so little, and thereby all the places will be filled: As practise will show. RULE 3. If the Aequation have but 5 dimensions, it must be brought up to 6 as followeth. Let it be + a ⁵ − ba ⁴ − dddaa + f4a − g ⁵ = o In stead of it wry + a ⁶ − ba ⁵ − d3aaa + f4aa − g5a = o And make e − q = a, and the thing will be effected which was desired: So if between two lines given b & c, and c > b, it be required to find 4 means, putting a for the lesser mean the Aequation will be + a ⁵ = = bbbbc, that is + a ⁵ = cb ⁴, And the places which are empty may be filled up by the second Rule: and brought to 6 dimensions (if yet it be not so) by the 3d. Rule. NOTE All Aequations whose dimensions are expressed 〈◊〉 numbers, can by no means be brought do●● 〈◊〉 fewer dimensions by any artifice that can be used: and for this reason the invention of any even number of means is much harder than to find an odd number of the like, so here where to find four means runs to an Aequation aaaaa = bbbbc, the Problem is absolutely Sursolid: but to find five means is as easy almost, as to find two, for the work brings us to the Aequation a ● = cb ⁵ which by the 5 Chapter may be brought down to the Equation. aaa = cdf. or to aaa = ddd if the Problem be plain, that is if it be c′ d″ f′″ otherwise by making df = gg it may be aaa = ggc. and the root a. found by a portion of a Parabola, as in the case of two means, Chap. 14. Lastly, for quinquisection, the Radius being unity, put the whole subtense b. The subtense of the fift part required a. Then + a ⁵ − 5 aaa + 5 a − b = o. As is demonstrated by Pitiscus, in his making the sins, Probl. 7. and shall here need no proof. It will require some labour (as may be seen by Chap. 15. Probl. 1.) to bring this Equation to the form required, for which purpose, first in stead of unity call Radius, r. And then it will be + a ⁵ − 5 rra ³ + 5 rrrra = 5 rrrrb, Afterward by Rule 3 it may be done. And when this is done, the rend●● 〈…〉 work, as well Construction as Demon 〈◊〉 〈◊〉, according to the method proposed by Des Cartes may be done also, but the doing will be both tedious and intricate. And therefore I shall no further prosecute the said Method, & for use, section of angles in general may be done by the Rule of False Positions, as the said Pitiscus made his Canon of Sines. But the Canon of Sines, the late Prince of Mathematicians VIETA, vouchsafes to call The Mathematical Canon. Nevertheless, if the industrious Reader desires more exactness (I mean in Theory) according to the former Method, or any other which he shall find better for his purpose, he may proceed at pleasure, A Rule for squaring binomiall surds, which should have been in Chap. 6. pag. 76. Multiply the quantity to which the sign √ belongs, into the square of the Coefficient, and the product is the square required. Example in Numbers. If the square of 3 √ 7 be required, multiply 7 into 9, the product is 63, the square required. Or, If the square of 4 √ 9 be demanded, 9 into 16 produceth 144, which is the square demanded: the like of all others. This needs no proof. See Page 77. And this shows that all such Surds are commensurable in power. Deo Gloria.